Download Studies in Quantum Information Theory

Studies in
Quantum Information Theory
Nicolas C. Menicucci
A Dissertation
Presented to the Faculty
of Princeton University
In Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of Physics
Adviser: Shivaji Sondhi
September 2008
c Copyright by Nicolas C. Menicucci, 2008. All rights reserved.
Abstract
Quantum information theory started as the backdrop for quantum computing and is often
considered only in relation to this technology, which is still in its infancy. But quantum
information theory is only partly about quantum computing. While much of the interest in
this field is spurred by the possible use of quantum computers for code breaking using fast
factoring algorithms, to a physicist interested in deeper issues, it presents an entirely new
set of questions based on an entirely different way of looking at the quantum world. This
thesis is an exploration of several topics in quantum information theory. But it is also more
than this. This thesis explores the new paradigm brought about by quantum information
theory—that of physics as the flow of information.
The thesis consists of three main parts. The first part describes my work on continuousvariable cluster states, a new platform for quantum computation. This begins with background material discussing classical and quantum computation and emphasizing the physical underpinnings of each, followed by a discussion of two recent unorthodox models of
quantum computation. These models are combined into an original proposal for quantum
computation using continuous-variable cluster states, including a proposed optical implementation. These are followed by a mathematical result radically simplifying the optical
construction. Subsequent work simplifies this connection even further and provides a constructive proposal for scalable generation of large-scale cluster states—necessary if there
is to be any hope of using this method in practical quantum computation. Experimental
implementation is currently underway by my collaborators at The University of Virginia.
The second part describes my work related to the physics of trapped ions, starting
with an overview of the basic theory of linear ion traps. Although ion traps are often
discussed in terms of their potential use for quantum computation, my work looks at their
potential for use as generic quantum systems over which the experimenter has exquisite
control and which can be used to simulate other quantum systems and also study generic
quantum phenomena. This is followed by a proposal for using a trapped ion as a timedependent harmonic oscillator—a quantum system that is common in theoretical literature
but of which few laboratory examples are known. A second project studies the way that
quantum fluctuations in the vibrational state of a chain of ions influence correlations in
optical measurements made on the ions.
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The final part looks at quantum information theory in a relativistic setting. An introduction discusses the interface between quantum information theory and relativity in
general, including the nonclassical notion of entanglement and the peculiar features of
curved-space quantum field theory. An original gedankenexperiment combines these ideas
and examines whether entanglement—a quantum information-theoretic concept and physical resource—can be used to distinguish universes of different curvature in a situation where
local measurements would show no difference.
These three parts are followed by a personal (and possibly controversial) conclusion,
which describes my fascination with—and ultimately my reason for pursuing—studies in
quantum information theory.
iv
Acknowledgments
My Ph.D. experience has been unique, challenging, productive, and also a whole lot of fun!
It’s amazing how many people significantly influence one’s life in five years—especially when
that life involves living in three cities that span two continents. Apologies in advance if,
despite my best efforts, I have ended up neglecting anyone.
There are several people that stand out vividly when I examine the path I’ve traveled
over the last five years. The first is Michael Nielsen. Michael has served in a number of
capacities for me: sponsor (of the entire program), project supervisor, co-author, mentor,
benefactor (providing research funding for me to travel and to present and promote my
work), and someone who is really interesting to talk to—about any topic related to physics
and many more beyond that, as well. Some of the best advice Michael ever gave me related
to my professional development. When I would run into a problem—lack of motivation, for
instance—Michael would listen to the problem, relate it to his own life, relate it to the lives
of other people he knows, analyze it from six different perspectives, summarize the findings,
and present a coherent strategy for addressing it. Every time I met with Michael I always
came away with a remarkable sense of confidence and calmness—regardless of how I felt
before. Michael has been an amazing mentor and role model—I hope to make him proud
throughout my career.
Gerard Milburn also stands out as someone who has provided a wealth of support
and expertise. Gerard took over as my local supervisor in Brisbane after Michael decided
to move to Perimeter Institute. The breadth of his knowledge is extensive and ranges
from the details and experimental development of quantum computer technology to the big
questions of quantum gravity. This full range of expertise—from the nitty gritty details of
the laboratory to theoretical questions that philosophers have been asking for thousands of
years that can now be formulated in the language of physics—makes Gerard a wellspring
of new ideas and helpful insights to any physics problem I brought to him, regardless of
topic. Gerard is also a person who is simply very pleasant to be around. He offered to let
me store some items at his house when I had to move out of my residential college, and he
hosted a number of parties and informal get-togethers for his students and colleagues. And
he was a good sport on his 50th birthday, when we threw him a surprise party.
When I look back on all of the decisions that led me down the path that I have chosen,
the one person who stands out—without a doubt—as the most influential in my career is
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Carl Caves. Carl’s technical skills as a physicist are rivaled only by his personal support of
his students—both past and present. Like Michael, Carl took me on in a visiting student
role—in this case, for my advanced project. But I have known Carl for almost a decade.
Beginning as a student in his electromagnetism class at The University of New Mexico
(UNM), Carl immediately showed himself to be both someone who students could learn
from and someone whose company would always be a welcome addition. Carl supervised
my honors thesis at UNM—a project that was ambitious but which demonstrated his faith
in me. It resulted in a publication in Physical Review Letters and, with Carl’s support,
ultimately to admission to a variety of prestigious graduate schools. I chose Princeton for
its emphasis on string theory and for the flexibility and support of the Physics Department,
and Carl was with me all the way through this process. When I decided to switch topics to
quantum information theory, Carl spoke with Michael (who was his former Ph.D. student)
on my behalf and helped me to bridge the gap between the path that I had chosen and the
new one on which I wished to embark. Carl’s influence was more subtle and personal, as
well. Before I moved to Australia, Carl hosted an Aussie movie night at his house, screening
“The Castle” and “The Dish” for his students in order to educate us about Australian humor
and culture. (If you haven’t seen them, I recommend them!) It’s the little things like this
that really make Carl stand out. I can say that Carl’s positive influence on my professional
life is second to none. While there are many others who have played a more active role in
the actual research that went into this Ph.D., it was Carl who was there at key junctions in
my life, and who helped me obtain the opportunity to succeed. I owe much of my current
success to his guidance and faith in me.
When it comes to the opportunity given to me to succeed on this unorthodox expedition,
there were a number of supporters in the Princeton Physics Department who encouraged
me to follow my dreams even if they involved risk or something not usually done. Paul
Steinhardt was the point man in this effort, presenting the Department (on the visiting day
I attended) as supportive and enabling of the best that the incoming students have to offer.
“The hardest thing about getting a Ph.D. from Princeton is getting in the door,” he told
the newly admitted students. This is not because it’s an easy process once you’re there—no.
It’s because even though it is challenging, everyone in the Department is expecting you to
succeed and is supportive of that success. Unlike some other universities, where admission
is broadened because of a need for teaching assistants (but without the intention to support
all of them through to the end), getting admitted to Princeton says, “We believe in you.”
Well, I have to say, I truly feel that the Physics Department has lived up to this promise. Of
particular mention in this regard is Bill Bialek, who from the day I met him made me feel
as if my success mattered to him on a personal level. When I decided to move away from
string theory, Bill was supportive and told me that I should study whatever I wanted to
study and that I should do it wherever was the best for me to do so. That is true support,
and it was very encouraging. Shivaji Sondhi echoed this sentiment, offering to serve as my
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official advisor from Princeton while I embarked on research in far away lands. This was
not without risk! After all, not even I knew I would be successful—but I believed I would
be, and Shivaji did too. And for that I am truly grateful. A huge thanks goes to Chiara
and Herman, who were supportive during their respective tenures as Director of Graduate
Studies, and I should also mention Laurel Lerner as being a huge help every step of the
way. I believe this truly is one of the most supportive physics departments in the world.
Naturally, the other half of the arrangement involves The University of Queensland
(UQ) Physics Department. In addition to Gerard and Michael, Halina Rubinsztein-Dunlop
gave the final go-ahead for the “visiting student” arrangement. She decided that having
me at UQ was a benefit to the department even though it meant I would not be paying
fees (i.e., tuition) to UQ. I am forever grateful for her support and the support of the
entire Physics Department, as well as the School of Physical Sciences. And finally, in the
same vein, I offer sincere thanks to the National Science Foundation Graduate Research
Fellowship (NSF GRF) Program and to the National Defense Science and Engineering
Graduate (NDSEG) Fellowship Program, who provided me with funding for the last four
years, and to Princeton University and Golden Key International Honour Society, which
provided funding for my first year of graduate studies. These fellowships gave me the
financial freedom necessary to achieve my goals.
Those are the big ones. Without them, I simply would not be where I am today. It
took a concerted effort of all of these people and organizations to encourage this work and
allow it to come to fruition. There are many others who played a more hands-on role both
in my professional and personal life that I should like to acknowledge, however. The first
group includes my co-authors, collaborators, and people with whom I have had very useful,
inspiring, and sometimes a bit off-the-wall discussions.
For the work on continuous-variable cluster states, particular mention goes to my coauthors Mile Gu, Michael Nielsen, Tim Ralph, Peter van Loock, Christian Weedbrook, Steve
Flammia, Olivier Pfister, Hussain Zaidi, Russell Bloomer, and Matthew Pysher. Discussions
were invaluable with Gerard Milburn, Andrew Doherty, Alexei Gilchrist, Guifre Vidal,
Andrew White, Mark de Burgh, Mark Dowling, Eric Cavalcanti, Henry Haselgrove, Austin
Lund, Ben Lanyon, and John Preskill. Particular mention should be made of Olivier Pfister
and Steve Flammia. These two world-class scientists have worked with me side-by-side
on a number of projects. Collaboration with Olivier began from our first meeting at the
Gordon Research Conference on Quantum Information Theory at Il Ciocco, Italy in 2006.
It was Olivier’s suggestion that we could generate continuous-variable cluster states from a
single OPO that spawned an entire research program in this area, beginning with our joint
work on it. An essential contributor to that project, as well as a long-time colleague on
a number of projects, Steve has been an indispensable collaborator as well as a wonderful
friend and housemate. Steve’s knowledge of mathematics is vast, and he is able to apply
vii
that knowledge to a wealth of physical problems. Our skills being complementary, when
Steve and I work on a problem, we form—as Steve would say—an incredible team.
The work with ion traps was facilitated by Gerard Milburn, Dave Kielpinski, Paul Alsing, Bill Unruh, John Preskill, Jeff Kimble. Gerard was my primary advisor, collaborator,
and co-author, while the others provided invaluable input on the project. For the relativistic
quantum information work, thanks goes especially and primarily to Greg Ver Steeg, who
ventured into the abyss with me and was an excellent and efficient collaborator—despite
being located a quarter of the way around the world and also a “lowly graduate student”
like myself. Additional thanks goes to John Preskill, Gerard Milburn, Carl Caves, Sean Carroll, and also the Caltech Institute for Quantum Information (IQI). Work on my advanced
project (not included in my thesis) involved collaborations with the UNM Information
Physics Group, especially Carl Caves and Steve Flammia, and also Seth Merkel, Aaron
Denney, Sergio Boixo, Bryan Eastin, Ivan Deutsch, and Joe Renes. I’d also like to thank
the Perimeter Institute for Theoretical Physics—especially Michele Mosca, Lee Smolin, and
Lucien Hardy—for their support of my attendence at the Quantum Foundations Summer
School and the Young Researchers Conference. In addition, the upcoming opportunity to
be a postdoc at Perimeter is one that I am very much looking forward to.
While I acknowledge these few professional colleagues by name, I feel—although it is
inadequate—that I must try to also acknowledged the countless other students and researchers who provided some particular piece of information at just the right time, likely on
more than one occasion, but whose individual contributions have fallen prey to the imperfections of human memory. Research is almost always a collaborative effort—to everyone
who helped make these research projects succeed, thank you!
Special thanks goes to Donna Sy, Rajat Ghosh, Eric Switzer, and Susannah Rutherglen
for letting me crash at their places during my visits (and also to whoever ends up hosting
me during my thesis defense). Mihail Amarie is applauded heartily for offering to help me
with the depositing process, and I am very much indebted to those other students who put
so much hard work into studying for prelims and generals way back in the day. Princeton
is a sleepy little town, but several important people made it a very fun place for me to live
and to visit. These include especially Donna Sy, Eric Switzer, Rajat Ghosh, Chris DeCoro,
Zafer Barutcuoglu. One very special person made Princeton a particularly delightful place
to be: Lisa Mruczek, an amazing woman full of beauty, love, and kindness, with whom I
had the privilege of sharing all kinds of adventures—from swing dancing, to the Poconos,
to our trip across the country—providing wonderful memories for years to come.
In New Mexico, on the personal side, I wish to thank Steve Flammia, Alex Theodorou,
Seth Merkel, Aaron Denney, and Journey Nolan, who were all my housemates at one time
or another. Very special thanks also goes to Aaron Cabral, who provided me with a place
to stay during numerous visits from Australia and was a wonderful bar hopping companion
and a truly loyal and supportive friend. In addition to these, Devon Hjelm, Evan Kias,
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Erin Husher, Mark Harris, Elizabeth Dao, and Erin Murrah provided much entertainment
and good times over many cups of coffee, bottles of beer, glasses of whiskey, and games
of pool. In something of a virtual world, I wish also to acknowledge almost daily chat
sessions with Gavin Mendel-Gleason and Praveen Sinha over IRC, spanning topics including
politics, economics, religion, biology, computer science, quantum physics, philosophy, and
our personal lives. Their constant input and intellectual stimulation have made me a better
person, as well as a better scientist, and I consider them to be great friends, as well.
My life in Brisbane certainly involved so many amazing people that I have no hope of
listing them all, so I will give a few that really stood out. First, I thank Jerome Haba,
Samantha Komaran, and Clare Gould, who all lived in the room adjacent to mine and
who were great fun to live with and party with. This is especially true of Jerome, who
was my bar hopping companion on many nights for more than a year and with whom
some amazing parties were hosted on Sir Fred Schonell Drive. I’d like to thank all of
my friends from Union College, who were lots of fun to hang out with and provided a
wonderful atmosphere in which to start my life in Brisbane. The UQ Physics Department
was, in many ways, my family. I’d like to thank Mark de Burgh, Mark Dowling, Mile Gu,
Katya Babourina, and Robert Pfeifer for being wonderful office mates, providing inspiring
discussions and ample entertainment. Matt Woolley, Charles Meaney, Aggie Branczyk, and
Sarah Midgley, among others, provided heaps of fun times in a social setting. On this front
though, Devon Biggerstaff and Mike Deceglie were by far my steadfast “partners in crime”
on many an eventful Friday and Saturday night. Some great memories were made in various
bars in Brisbane over a jug (of beer) or three. The entire Queensland University Musical
Society (QUMS) has been a continuing source of fun, entertainment, shenanigans, and—of
course—music over the past two years. In particular, Nelson Vermeer has been one of my
best friends, going so far as to invite me to his family’s house at Christmas last year while
I was far away from my own. Through the choir, I met a huge number of amazing people,
including members of other university choirs around Australia, randoms in Byron Bay and
the Woodford Folk Festival, and plenty of others in Brisbane. It was also through the choir
that I met a very special and truly awesome person: Kristy “Lady Mirlette” Mannell, who
(despite having to put up with my repeated unavailability due to a certain Ph.D. thesis to
be completed) remained forever fun, intelligent, and beautiful—a truly magnificent addition
to my life and someone who will remain in a special place in my heart for years to come.
I shall close these lengthy notes with a heartfelt thanks to my family. My mother and
father, Barbara and David Menicucci, have always been and to this day remain my biggest
fans and loudest supporters. It goes without saying that without them I wouldn’t be here.
But they have provided me with much more than physical form—they raised me to revere
learning, to think independently, and to use my talents to pursue my goals. In addition, my
brother Anthony has always been a source of fun and entertainment for myself and everyone
else at family gatherings. Many others on the Menicucci side of the family—notably Emma,
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Paul, and Kirk and Kathy Meadows—remain stalwart supporters my success and also of
me, personally.
Both of my grandfathers passed away during my Ph.D., and they deserve a special mentioning because they were both extremely supportive of learning and intellectual pursuits.
Robert Dinegar (“Grandpa”) was a chemistry professor at the Los Alamos branch of The
University of New Mexico. A passionate but very practical man, as well as an Anglican
priest, Grandpa loved the sciences. In large part I feel I am continuing in his footsteps with
my own chosen career. Charlie Menicucci (“Granddad”) was more modestly educated, but
his passion for learning was just as strong. He worked at a service station for most of his life
in Albuquerque and was a local expert on cars for many years even after retirement. He led
a modest life, providing steadily and sufficiently for his family so that, as his father “Nonno”
Julio Menicucci would say, “each generation may do better than the previous one.” Nonno
Julio wisely chose not to define “better,” but I’d like to think both he and his son Charlie
would be proud. As a final point, I would also like to honor the memory of my beloved
childhood pet, Moe King Cat (“Mokey”), who passed away peacefully while I was on the
other side of the world. The only friend from back home who I could never get to talk to
me on the phone, he is also dearly missed.
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Contents
Abstract
iii
Acknowledgments
v
Contents
xi
List of Figures
xv
List of Tables
xvii
Introduction
1
1 More Than Just Breaking Codes
2
I
1.1
A New Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Continuous-Variable Cluster States
2 Novel Approaches to Quantum Computation
7
8
2.1
Circuit Model of Classical Computation . . . . . . . . . . . . . . . . . . . .
10
2.2
Circuit Model of Quantum Computation . . . . . . . . . . . . . . . . . . . .
12
2.2.1
Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.2
Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.3
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
One-Way Quantum Computation Using Cluster States . . . . . . . . . . . .
17
2.3.1
Quantum Wires
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.2
Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.3
Universal Cluster States . . . . . . . . . . . . . . . . . . . . . . . . .
22
Quantum Computation with Continuous Variables . . . . . . . . . . . . . .
24
2.4.1
Qudits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4.2
Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
2.4
xi
2.4.3
Continuous-Variable Cluster States . . . . . . . . . . . . . . . . . . .
3 Universal Quantum Computation with CV Cluster States
28
29
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Continuous-Variable Cluster States . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
Optical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.4
Errors Due to Finite Squeezing . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.5
Experimental Proposal for Cluster-Based Error Reduction . . . . . . . . . .
34
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4 Ultracompact Generation of Continuous-Variable Cluster States
37
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2
CV Clusters and Multi-Mode Squeezing Hamiltonians . . . . . . . . . . . .
39
4.3
Experimental Proposal: Square-Graph CV Cluster State . . . . . . . . . . .
44
4.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5 Entangling the Optical Frequency Comb: Simultaneously Generating Many
Small CV Cluster States
47
5.1
Multipartite entanglement in the optical frequency comb . . . . . . . . . . .
47
5.2
H (Hamiltonian)-graph states: physical description . . . . . . . . . . . . . .
48
5.3
Square-cluster OPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.4
Multiple square cluster states from a single OPO . . . . . . . . . . . . . . .
51
5.4.1
Principle and first experimental implementation . . . . . . . . . . .
51
5.4.2
Second experimental implementation . . . . . . . . . . . . . . . . . .
52
5.4.3
Third experimental implementation . . . . . . . . . . . . . . . . . .
54
5.5
Simplified relationship between H-graphs and cluster-state graphs . . . . .
55
5.6
Simultaneously generating multiple copies of a CV cluster state . . . . . . .
56
5.7
Simultaneous generation of 2 × 2 and 2 × 3 cluster states . . . . . . . . . . .
57
5.8
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6 One-Way Quantum Computing in the Optical Frequency Comb
60
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
6.2
CV cluster states from a single OPO . . . . . . . . . . . . . . . . . . . . . .
61
6.3
Single Mode-Universal CV Cluster State . . . . . . . . . . . . . . . . . . . .
62
6.4
QC-Universal CV Cluster State . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.5
Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
71
6.6
Finite squeezing and CV fault tolerance . . . . . . . . . . . . . . . . . . . .
73
6.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
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II
Trapped Ion Physics
75
7 Trapping and Controlling Ions
76
7.1
Trapping a Single Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
7.2
Laser Coupling and Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
7.3
Multiple Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
7.4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
8 A Single Trapped Ion as a Time-Dependent Harmonic Oscillator
88
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
8.2
General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
8.3
Exponential Chirping
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
8.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
9 Spatial Correlation Functions for the Collective Degrees of Freedom of
Many Trapped Ions
98
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
9.2
Spatial Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
99
9.3
Measurement of Ion Trap Spatial Correlations . . . . . . . . . . . . . . . . .
101
9.3.1
Normal modes of vibration . . . . . . . . . . . . . . . . . . . . . . .
101
9.3.2
Laser-induced coupling of vibrational and electronic states . . . . . .
103
9.3.3
Excitation probabilities and correlation functions . . . . . . . . . . .
105
Excitation Probability Calculations for General States . . . . . . . . . . . .
107
9.4.1
Long-time interaction . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Evaluation for Gaussian States . . . . . . . . . . . . . . . . . . . . . . . . .
110
9.5.1
Two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
9.5.2
Probabilities in terms of the covariance matrix . . . . . . . . . . . .
113
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
9.6.1
Thermal state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
9.6.2
Uniformly squeezed normal modes . . . . . . . . . . . . . . . . . . .
117
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
9.A Appendix: Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
9.A.1 Gaussian States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
9.4
9.5
9.6
9.7
III
Entanglement in Curved Spacetime
10 Relativistic Quantum Information Theory
124
125
10.1 The Fall of Local Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
10.1.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
10.1.2 Bell Inequalities and Hidden-Variable Models . . . . . . . . . . . . .
127
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10.1.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
10.2 Quantum Information Theory in Curved Spacetime . . . . . . . . . . . . . .
133
10.2.1 Particles? What particles? . . . . . . . . . . . . . . . . . . . . . . . .
134
10.2.2 Introducing Curvature . . . . . . . . . . . . . . . . . . . . . . . . . .
134
10.2.3 Horizons, Radiation, and Quantum Information Theory . . . . . . .
135
11 Entangling Power of an Expanding Universe
138
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
11.2 Two Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
11.3 Entanglement Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
11.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
Conclusion
147
12 Why Quantum Information Theory?
148
Appendix
153
A Quantum Optics Cheat Sheet
154
A.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 Quantization of the electromagnetic field
154
. . . . . . . . . . . . . . .
154
A.1.2 Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
A.1.3 Gaussian states, Gaussian operations, and the Heisenberg picture . .
156
A.1.4 Single- and multi-mode-squeezed states . . . . . . . . . . . . . . . .
158
A.2 Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
158
A.2.1 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
A.2.2 Squeezing—nonlinear media . . . . . . . . . . . . . . . . . . . . . . .
159
A.2.3 Optical parametric oscillator (OPO) . . . . . . . . . . . . . . . . . .
160
Bibliography
163
xiv
List of Figures
2.1
Computation as a physical process . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Half-adder circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Square-lattice cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.1
Experimental schematic for single-OPO cluster state generation . . . . . . .
44
4.2
Single-OPO generation of a square-graph continuous-variable cluster state .
45
5.1
Pairwise entanglement from a single pump mode . . . . . . . . . . . . . . .
48
5.2
Four squeezing interactions from a bimodal pump . . . . . . . . . . . . . . .
50
5.3
Multiple square cluster states from a bimodal pump . . . . . . . . . . . . .
52
5.4
Multiple squares from a single-mode pump using polarization . . . . . . . .
53
5.5
Multiple squares from a single-mode pump using polarization (alternate) . .
54
5.6
Cubic cluster-state graph . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.1
Hankel shorthand and pump specification . . . . . . . . . . . . . . . . . . .
63
6.2
Interpretation of a matrix-weighted edge . . . . . . . . . . . . . . . . . . . .
64
6.3
Matrix-valued weights and supergraphs . . . . . . . . . . . . . . . . . . . .
65
6.4
Four-color solution to the geometric orthogonality conditions . . . . . . . .
67
6.5
Circulant embedding of a twisted toroidal lattice . . . . . . . . . . . . . . .
69
6.6
Toroidal lattice supergraph and underlying graph structure . . . . . . . . .
70
6.7
Unrolling the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
7.1
Schematic of a linear ion trap . . . . . . . . . . . . . . . . . . . . . . . . . .
77
7.2
Laser-induced electronic transitions in a trapped ion . . . . . . . . . . . . .
82
7.3
Measurement of an ion’s electronic state by fluorescent shelving . . . . . . .
83
7.4
Equilibrium positions for multiple ions in a linear trap . . . . . . . . . . . .
84
9.1
An illustration of the linear ion array with N ions . . . . . . . . . . . . . .
107
9.2
Correlation functions for the normal modes of 10 ions . . . . . . . . . . . .
118
10.1 Spin axis anti-correlation for two classical tops . . . . . . . . . . . . . . . .
128
11.1 Entanglement profile for detector pairs in several universes . . . . . . . . . .
145
xv
A.1 Examples of Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
A.2 Schematic of homodyne detection . . . . . . . . . . . . . . . . . . . . . . . .
159
A.3 Schematic of an optical parametric oscillator (OPO) . . . . . . . . . . . . .
160
xvi
List of Tables
1.1
Thesis chapters with associated references to my coauthored papers. . . . .
xvii
6
Introduction
1
Chapter 1
More Than Just Breaking Codes
Green light illuminates the cold walls like the flicker of candlelight, only much more ominous,
as if reflecting the inner workings of a mad scientist’s brain. The laser pulses unendingly
against the dark backdrop of high-tech optics, as the photodetectors click rapid-fire in what
sounds like a cross between Morse code and a Geiger counter at the Trinity testing site.
“They’re getting what they deserve.” The judgment pierces the eerie silence otherwise
disturbed only by the whir of fans and the clicks of the optical components spiking on and
off, selectively measuring—and in such a way, manipulating—a light beam made to do far
more than just light up the room. “If they’re stupid enough to still use RSA, then they
deserve to get hacked,” clarifies the defiant voice now clearly in front of a dimly-lit flat-panel
monitor. The faint light from the laser lights up cords running from detectors to computers,
back to detectors, entangled into a huge, haphazard braid-like mass of control.
The minutes pass by without further interruption, save for a few taps on an old-style
keyboard attached to an old-style dual-core computers—once the powerhouses of the desktop world now relegated to the status of humble workhorses for a much more powerful
processor: the quantum cluster-computing core, or QCCC. The QCCC is not your typical microprocessor. It exploits quantum properties of light to compute in an entirely new
way, sidestepping the restrictions of ordinary computing that make public-key encryption
“secure.” The room darkens, and the clicking ceases. A cryptic message appears on the
monitor:
26B040CDD04126B3513DA80A4029478E9DB41553E81097257E102EA93031F4AA
D57C0992C0C07F47266C46917E108EB53B608F9B355B3F46A99ECD5F0D09F279
799B396C63909C1DCEC7DED27F3B28291376A8B2215D6DA9B76EF04052712C4D
9F06E555210945E39271E8609D224CFA672E5F75BF8A3AFED89F152737932987
Use of the QCCC reduced the time needed to crack this 1024-bit private key to mere hours,
where it would have taken decades with ordinary hardware. “Time to see what you’re
hiding,” muses the anonymous hacker. . . .
2
1.1
A New Paradigm
The scenario just described is fictional. Working quantum computers do not yet exist on the
scale large enough to break modern encryption protocols, which rely on keys around 1024
bits in size for very secure systems; current state-of-the-art quantum computing hardware
can factor 15, a four-bit number. But quantum information theory is only partly about
quantum computing. It is no secret that much of the interest in quantum information
theory research is spurred by the possible use of quantum computation for code breaking
via fast factoring algorithms. But to a physicist interested in deeper issues, it presents an
entirely new set of questions based on an entirely different way of looking at the quantum
world.
Physics, for centuries, has focussed on matter and energy—“stuff”—and the way it
interacts with other “stuff.” Mathematical laws have been formulated to describe the interactions of the “stuff.” Quantum information theory has begun to change that entire
paradigm. Quantum information theory started as the backdrop for quantum computing.
Attempts to develop quantum computing technology led to questions of what resources
give quantum computers their power—what differentiates them fundamentally from classical computers? Entanglement arises as a natural answer to this question, since entangled
states cannot be described classically (see Chapter 10). Over time, though, entanglement
comes to be seen as less of an abstract property of quantum states and instead as a physical
resource (see Chapter 2). Quantum information theory begins to examine the quantum
world in terms of the way it can be used to process information (beyond just computation).
As it becomes clear that there is a richness to the states and dynamics of quantum theory
independent of the physical realization, quantum information theory starts to include the
study of state preparation, control, and measurement in a variety of systems (including
ion traps—see Chapter 7). The creation, manipulation, and general study of nonclassical
states takes on a life of its own within quantum information theory and leads finally to big
questions about the foundations of quantum mechanics. What is a quantum state? How
do we address the “measurement problem” associated with wavefunction collapse? There
are many answers to these questions, and quantum information theory brings a unique
perspective, which also raises questions of its relevance to other areas of physics, including
especially relativity theory (see Chapters 10, 11, and 12).
I encourage you, the reader, to bear these questions in mind while reading this thesis.
They provide a unifying theme to the apparently disparate topics, and they will be revisited
in Chapter 12 in light of the research described in the intervening chapters.
1.2
Structure of the Thesis
The original research presented in this thesis is collected from the papers I have published
over the past three years. For reference, and to give my coathors due credit on our joint
3
work, Table 1.1 lists these papers alongside the thesis chapter that includes that work. My
work covers three main areas, organized as the three main parts of the thesis:
• Part I, consisting of Chapters 2–6, describes my work on continuous-variable cluster
states, a new platform for quantum computation.1 Chapter 2 provides the background
material, discussing classical and quantum computation and emphasizing the physical
underpinnings of each, followed by a discussion of two recent unorthodox models of
quantum computation. These models are combined in Chapter 3 into an original
proposal for quantum computation using this method, including a proposed optical
implementation. Chapter 4 proves a mathematical result radically simplifying the
optical construction from Chapter 3, although work remains to be done to show
that the complexity of the new method is manageable. Chapters 5 and 6 do exactly
this, simplifying the mathematical connection built in Chapter 4 and providing a
constructive proposal for scalable generation of large-scale cluster states—necessary
if there is to be any hope of using this method in practical quantum computation.
• Part II consists of Chapters 7–9 and describes my work related to the physics of
trapped ions. Chapter 7 presents an overview of the basic theory of linear ion traps.
Although ion traps are often discussed in terms of their potential use for quantum
computation, my work looks at their potential for use as generic quantum systems over
which the experimenter has exquisite control and which can be used to study generic
quantum phenomena. Chapter 8 proposes using a trapped ion as a time-dependent
harmonic oscillator—a quantum system that is common in theoretical literature but
of which few laboratory examples are known. Chapter 9 studies the way that quantum
fluctuations in the vibrational state of a chain of ions influence correlations in optical
measurements made on the ions.
• Part III is the final part and covers Chapters 10 and 11. The introductory chapter,
Chapter 10, discusses the interface between quantum information theory and relativity, including the nonclassical notion of entanglement and the peculiar features
of curved-space quantum field theory. Chapter 11 combines these ideas and examines whether entanglement—a quantum information-theoretic concept and physical
resource—can be used to distinguish universes of different curvature in a situation
where local measurements would show no difference.
As should be clear from this list, each part begins with a chapter containing a technical
overview of the concepts necessary to understand the original research presented in subsequent chapter(s). With such a diverse selection of topics, a comprehensive introduction
to all of them could cover several volumes and would be beyond my purpose—which is to
provide appropriate background for my work. With this purpose clearly in mind, while still
1
This work forms the basis for the fictional “quantum cluster-compuuting core (QCCC)” in the story
above.
4
wishing to paint a comprehensive picture, many interesting and exciting research avenues
not directly related to my research are left to citations of relevant books, review articles,
and other literature. The three parts are followed by a personal (and possibly controversial)
conclusion in Chapter 12, which describes my fascination with—and ultimately my reason
for pursuing—studies in quantum information theory.
5
3
4
5
6
8
9
11
Part I: Continuous-Variable Cluster States
Nicolas C. Menicucci, Peter van Loock, Mile Gu, Christian Weedbrook, Timothy C. Ralph, and Michael A. Nielsen, “Universal quantum computation with
continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006) [1]
Nicolas C. Menicucci, Steven T. Flammia, Hussain Zaidi, and Olivier Pfister,
“Ultracompact generation of continuous-variable cluster states,” Phys. Rev. A 76,
010302(R) (2007) [2]
Hussain. Zaidi, Nicolas C. Menicucci, Steven T. Flammia, Russell Bloomer,
Matthew Pysher, and Olivier Pfister, “Entangling the optical frequency comb: simultaneous generation of multiple 2 × 2 and 2 × 3 continuous-variable cluster states
in a single optical parametric oscillator,” Laser Phys. 18, 659 (2008) [3]
Nicolas C. Menicucci, Steven T. Flammia, and Olivier Pfister, “One-way quantum computation in the optical frequency comb,” arXiv:0804.4468 [quant-ph]
(2008) [4]
Part II: Trapped Ion Physics
Nicolas C. Menicucci and Gerard J. Milburn, “Single trapped ion as a timedependent harmonic oscillator,” Phys. Rev. A 76, 052105 (2007) [5]
Nicolas C. Menicucci and Gerard J. Milburn, “Spatial correlation functions for
the collective degrees of freedom of many trapped ions,” arxiv:0807.3500 [quant-ph]
(2008) [6]
Part III: Entanglement in Curved Spacetime
Greg L. Ver Steeg and Nicolas C. Menicucci, “Entangling power of an expanding
universe,” arXiv:0711.3066 [quant-ph] (2007) [7]
Table 1.1: Thesis chapters with associated references to my coauthored papers.
6
Part I
Continuous-Variable Cluster States
7
Chapter 2
Novel Approaches to Quantum
Computation
Computation of any sort is simply the controlled manipulation of information. This in
itself is an abstract concept and can be formulated entirely mathematically. At the heart
of the notion of computation, though, is a sense that a physical process must be carried out
that corresponds to this manipulation—or processing—of information [8]. This means that
computable operations are limited by what we can do in the world, and this, in turn, is
limited by what we know about the world.
Before the discovery of quantum mechanics in the early twentieth century, classical
Newtonian physics was the best model available for the processes that could take place
in the world. These processes allow some things, for instance, the perfect “cloning” (i.e.,
duplication) of information, while forbidding others, such as the transmission of information
faster than the speed of light. The laws of nature restricted the laws of computation.
Quantum mechanics overturned some of those laws, imposed others, and left some unchanged. Perfect cloning of quantum states is, in general, impossible [9, 10], so information
stored in a quantum state cannot always be duplicated perfectly. On the other hand, quantum entanglement allows for correlations to exist (e.g., between pairs of particles) that are
stronger than any classical model of reality would allow [11, 12] (see also Chapter 10).
The best known classical algorithms for unstructured database search and integer factoring
are outperformed in efficiency by their best known quantum counterparts—respectively,
Grover’s algorithm [13, 14] and Shor’s algorithm [15]. Still, both we and the computational
machines we build are subject to the universal speed limit of c, which limits how fast information can be processed.1 While it is still unclear what property gives these efficient
1
Some people will argue that quantum mechanics is “nonlocal” and breaks this speed limit at the ontological level (the de Broglie-Bohm interpretation [16, 17, 18] being the quintessential example). Regardless
of what you choose as your favorite interpretation of quantum mechanics, it is well established that fasterthan-light signaling is always prohibited in the quantum context [19], and this is the relevant point for
information processing. See also Chapter 10.
8
abstract:
input
(information)
computation
output
(information)
physical:
initial state of a
physical system
physical process
final state of a
physical system
Figure 2.1: Computation as a physical process. All computation consists of an abstract input
being processed and transformed into an abstract output. The entire process can be modeled as
information flow (top line), but the underlying model is that computation is a physical process
that evolves an initial state of a physical system to a final state. Solid arrows indicate the actual
process by which computation occurs. Dotted arrows indicate the abstracted short-cut that is
extracted from the physical process and thus whose rules are bound by our understanding of
the laws of physics.
quantum algorithms an advantage2 over their best known classical ounterparts [20], the
undeniable fact remains that the discovery of quantum mechanics has changed the rules of
computation.
Any physical instance of computation—whether classical or quantum—can be modeled
as the process shown in Figure 2.1. Although it is often convenient to abstract away the
physical steps into the abstract processing of information, the essential difference between
classical and quantum computation has to do with the underlying physicality of computation. Classical computation can be implemented on systems that obey only the laws of
classical physics (a subset of quantum physics). Quantum computation, on the other hand,
can only be implemented on quantum systems.
There are several caveats to this claim that will distract from the main point if not at
least minimally addressed, so I will briefly do so here. First, quantum computation can
be simulated using classical computation, for instance using a computer algebra package
on your desktop computer to evolve the amplitudes of the quantum wave function and
calculate expectation values. This does not change the definition of quantum computation
as requiring a quantum world—simulated or not—for its implementation. Second, issues of
efficiency, which I briefly mentioned earlier, are the main motivation for research in (and
funding for) quantum computation. It is widely believed that quantum computers are more
efficient than classical computers for certain tasks in terms of the resources required (running
time, apparatus size, number of components, etc.) as the size of the problem increases [22].
For instance, factoring a two-digit number like 15 is easy—it’s just 3 times 5. Publickey encryption [23], however, relies on the belief—although unproven—that factoring huge
numbers is “hard” (inefficient) on a classical computer—that is, the time required to do so
2
It still has not been rigorously proven that quantum computers are actually more efficient than classical
computers, although it is widely believed to be true [20, 21, 22].
9
scales exponentially in the number of digits in the number to be factored [20, 24]. A quantum
computer can factor a number “easily” (efficiently) using Shor’s algorithm [15, 20, 25] in a
time that scales polynomial in the number of digits. Connecting the two concepts, it is also
believed [26] that classical computation cannot efficiently simulate quantum computation
in general. This means that to take advantage of the efficiency of Shor’s algorithm, it is
strongly believed that we must use “quantum hardware.” The current state of the art in
quantum hardware can factor the number 15 [27, 28]. Clearly this technology is still in its
infancy.
The literature and range of topics in quantum computation from the computer science
angle is vast. Still other computing-related novelties from the quantum world, such as
quantum key distribution—the main component of quantum cryptography—will also not
be touched on here. A good place to start for a comprehensive overview of quantum computation from both the physical side and the side of computer science is Quantum Information
and Quantum Computation by Nielsen and Chuang [22], as well as the references therein.
This description has been sufficiently broad-brush in order to paint the background
that I will use to illustrate two novel approaches to implementing quantum computation in
the physical world. First, though, I will describe the standard “circuit” model of classical
quantum computation and show how they are naturally analogous. Following that, I will
describe a recently invented method called one-way quantum computation, which makes
use of a highly entangled resource state called a cluster state, and finish with a discussion
of using continuous variables for quantum computation. This will provide the necessary
background for the following four chapters covering my work on continuous-variable cluster
states.
2.1
Circuit Model of Classical Computation
There are a number of equivalent models of classical computation (Turing machine [8],
circuit model [22], λ-calculus [29], etc.), but the one that is most useful in generalizing to
quantum computation is the circuit model. Here I briefly review the classical circuit model
and describe its most common quantum analog. The focus is on the concept of universality,
including the notion of a universal gate set, in the classical case and (later) in the quantum
case. A full description of the circuit model and universality for both classical and quantum
computation can be found in Ref. [22].
A set—or “toolbox”—of computing resources is said to be universal if any computable
function can be implemented using only elements from that set (in finite quantities). In the
classical circuit model of computation, information is represented by bits, systems with two
states that are labeled abstractly by ‘0’ and ‘1’. Computable functions therefore include
any Boolean function f : {0, 1}N → {0, 1}M whose truth table can be given explicitly,
10
for all positive integers N and M .3 The beauty of universality is that any such Boolean
function may be constructed from a universal set of simple Boolean functions called logic
gates (and, or, not, etc.). These logic gates have their usual definitions, e.g., x and y
outputs 1 only if x and y are both 1 (otherwise, 0). The ability to compose these logic
gates in any fashion and to prepare auxiliary “ancilla” bits with known initial values is
assumed when we say that {and, or, not} forms a universal gate set—that is, a set of
simple functions from which any Boolean function f can be constructed. Universal gate
sets are not unique. In fact, the one-element set {nand}, where nand is equivalent to and
followed by not, is universal (along with appropriate use of ancillas), since and, or, and
not can all be constructed from it. This method of using the gates of one set to implement
those of a different, known-universal set is a useful way to prove universality.
At this point, the reader should feel somewhat uneasy since this chapter began by driving
home the point that the laws of nature affect the laws of computation while the discussion
of universality above was wholly abstract. This is a critical point, and in fact, it marks the
juncture between classical and quantum computation. The classical model of computation
described above has certain assumptions built into it, the most important of which being
that classical logic gates need only classical physics for their physical implementation. Modern personal computers, which make extensive use of nanotechnology in all aspects of data
processing (e.g., CPU, motherboard, on-board memory) and data storage (e.g., hard drive,
optical drive), necessarily have to contend with (and often rely upon!) quantum phenomena
for their design and implementation.4 As such, while they are the most familiar examples
of classical computation, they are not the most instructive for our purpose.
The use of transistors and other nanotechnological devices has enabled the radical miniaturization of computers over the last couple of decades, but in principle, everything that
can be done on your personal computer can be done using mechanical devices like pistons,
levers, and cogwheels, which can be wholly described using classical physics. Using the universality result from above, if we can construct physical systems that implement the logic
gates in a universal set—e.g., and, or, and not, or even just nand alone—as long as we
can also compose them in any arbitrary functional fashion, we can construct a machine that
can—at least in principle—do any calculation that can be done on a desktop computer. The
functional composition of logic gates lends itself to a diagrammatic representation, called
a circuit (for which the circuit model is so named). A simple circuit called a half-adder is
illustrated in Fig. 2.2.
3
In classical logic (the usual logic that we learned in school), there exist functions whose action can be
specified completely but whose truth table cannot be given explicitly. An example is the halting function,
which takes a program (encoded as a number) as input and returns 1 if it will eventually halt when executed
and 0 if it will never halt. This function is not computable. See Ref. [22] for more details.
4
For instance, giant magnetoresistance [30, 31]—a quantum effect—was a breakthrough in nanotechnology
that allowed for the spectacular miniaturization of modern hard drive technology and earned its discoverers,
Albert Fert and Peter Grünberg, the 2007 Nobel Prize in Physics.
11
Figure 2.2: Half-adder circuit. This simple classical circuit implements the base-two addition
of two input bits. The output is x xor y, implemented as (x or y) and not (x and y), with
a carry bit set to x and y.
The power of the basic logic gates lies in their simplicity. They are so simple, in fact, that
they can be implemented using children’s construction toys, such as LEGOs.5 Connecting
together such simple, physical circuit elements allows for construction of the half-adder
of Fig. 2.2.6 A more complicated classical computing machine, the difference engine [32]
of Charles Babbage, is (and was designed to be) entirely mechanical and can be used to
evaluate polynomials near a given point. This can be constructed out of LEGOs, as well.7
While no one would reasonably expect to do word processing or signal analysis on a LEGO
computer, there is no barrier in principle to doing so. No more than a universal gate set
and classical physics are needed to perform classical computation.
The crucial elements of the classical circuit model that will transfer over to the quantum
case are universality and realizability. Universality refers to the existence of a universal gate
set—a “toolbox” that can be used to implement any computable function. Realizability is
the property that the gates in the universal set, as well as connections between them and
appropriate ancilla, can be implemented physically in the real world. Let’s see what changes
when we account for the quantum nature of reality.
2.2
Circuit Model of Quantum Computation
In generalizing to quantum computation, we will start with physical quantum systems first
and then discuss what abstract operations allow for universality. This presentation will
be sufficient for its purpose, which is to introduce the notion of universality in quantum
computation, but many well-document details, caveats, and their discussions will be omitted
in the interest of clarity. Reference [22] has a thorough description of quantum universality,
including many of these nuances.
5
At the time of this writing, photos, descriptions, and instructions for building LEGO logic gates can be
found at http://goldfish.ikaruga.co.uk/logic.html .
6
At the time of this writing, a video of one industrious youngster implementing this using circuit elements
made from K’NEX toys can be found at http://www.youtube.com/watch?v=3vXlQZvS-nM .
7
http://acarol.woz.org/
12
2.2.1
Universality
As discussed previously, it is believed that quantum computation can only be efficiently
implemented on “quantum hardware.” We therefore start by requiring two-state quantum
systems to be the information carriers in our quantum computer—as opposed to classical
bits. These “quantum bits”, or qubits, can carry classical information just as well as ordinary bits can. That is, they can be placed in one of two mutually exclusive (orthogonal)
states, labeled |0i and |1i. These states form what is called the computational basis, which
provides a basis for projective measurement at the end of the computation (to obtain a
classical result). But qubits can do much more than this. Quantum mechanics allows for
superpositions within each qubit, e.g., |ψi1 =
mutliple qubits, e.g., |ψi123 =
√1 (|001i
2
√1 (|0i + i |1i),
2
and entanglement [22] between
− |110i). Entanglement is an inherently quantum
phenomenon, producing correlations in the outcomes of repeated measurements stronger
than can be modeled by classical physics [11, 12], much to the dismay of “realists” like
Einstein, Podolsky, and Rosen (EPR), who first noticed this property of certain quantum
states [33]. (Entanglement will be discussed in more detail in Chapter 10.) Despite—or
perhaps, as a result of—challenging our classical intuition, entanglement is now believed by
many to be an essential feature of quantum computation [20] (although this claim is still
controversial [34, 35]).
Real-world quantum systems evolve according to Hamiltonians, which generate unitary
time evolution [22]. Since our information carrying system undergoing such evolution is a
set of qubits, we will define the set of quantum-computable functions to include any possible
evolution of that system—that is, SU(2N ), the set of all unitary operators on N qubits,
for all positive integer N . Comparing this with the classical definition of computable function as a Boolean function f as defined above, we note several things: (1) being unitary,
quantum-computable functions are reversible, while classically computable functions need
not be; (2) every classically computable function f that is also reversible has a (nonunique)
corresponding quantum implementation Uf ; and (3) if f is not reversible, ancilla bits can
be used to make it so by distinguishing between identical outputs that are the result of
distinct inputs. Combining these results, all classically computable functions can be implemented on a quantum computer, and thus quantum computation is universal for classical
computation. Interestingly, the converse is true as well, since as discussed earlier in the
chapter, a classical computer can simulate the unitary evolution of qubits, although (it is
believed) not always in an efficient fashion.
Universality in the quantum context is thus reduced to the ability to implement any
desired unitary operation on N qubits. Since we want to generate unitaries, a universal
gate set for quantum computing is a set of simple unitaries that can be used to implement
any member of SU(2N ) for arbitrary N . One such set is SU(2) ∪ {CX }: the set of all
single-qubit unitaries plus the controlled-not operation (denoted by CX ). The CX gate is
a two-qubit unitary that exchanges |0i and |1i on the ‘target’ qubit if the ‘control’ qubit is
13
in the state |1i and does nothing if the qubit is in the state |0i [22]. Being a permutation,
this semi-classical description of the action of the CX gate is sufficient to define its unitary
action on any state. More concretely, using the ordering (control ⊗ target), the action of
CX on the four computational basis states is |00i → |00i, |01i → |01i, |10i → |11i, and
|11i → |10i. The single-qubit gates form a continuous manifold, but I will define a few of
the most important ones here, in the standard (computational) basis {|0i , |1i}:
(Pauli gates)
(Hadamard gate)
(phase gate)
(π/8 gate)
X=
0 1
T =
,
1 0
1
1
H=√
2
S=
!
1 0
1
i
0
,
1
Z=
0
!
0 −1
,
,
(2.2)
,
0
0 eiπ/4
(2.1)
!
1 −1
!
0 i
1
Y =
!
0 −i
(2.3)
!
iπ/8
=e
!
e−iπ/8
0
0
eiπ/8
.
(2.4)
Notice that X acts as a not gate in the computational basis, swapping |0i ↔ |1i. This is
why CX is used to denote controlled-not: it acts as a conditionally applied X gate.
One can control any unitary operation, however. Another very common gate is the
controlled-Z gate, denoted CZ , which applies Z to the target qubit if the control qubit
is |1i and does nothing if the control is |0i. It has the misfortune of being commonly
called the “controlled-phase” gate, even though the operation being controlled is the Z
gate (a Pauli operator) and not the phase gate S. For completeness, here are the matrix
representations of the CX and CZ gates in the standard basis {|00i , |01i , |10i , |11i}:
(controlled gates)
CX

1

0
=
0

0

0 0 0

1 0 0
 ,
0 0 1

0 1 0

1 0 0
0


0

0

0 0 0 −1

0 1 0
CZ = 
0 0 1

(2.5)
Notice that, unlike the CX gate, the CZ gate is symmetric with respect to control and target
since the only basis state that gets modified at all is |11i, which acquires a phase of −1.
Appropriate combinations of single-qubit unitaries on individual qubits and CX gates
between pairs of qubits allows for the exact implementation of any unitary on an arbitrary
number N of qubits [22]. Therefore, SU(2) ∪ {CX } constitutes a universal gate set for
quantum computation. Also notice that since Z = HXH, then CZ = (1 ⊗ H)CX (1 ⊗ H),
which means the CZ gate can replace the CX gate as the required two-qubit operation,
making SU(2) ∪ {CZ } also a universal set. More generalizations are possible, but we will
not need them.
14
This universality construction above is exact, meaning that SU(2)∪{CX } can be used to
implement any U ∈ SU(2N ) exactly. The price we pay for this accuracy is two-fold: (1) we
must be able to implement an infinite set of gates, and more importantly, (2) a fault-tolerant
implementation for all of these gates is not known to exist. A quantum computation is fault
tolerant if it is performed in a manner that is resistant to errors. A fault-tolerant threshold
is an error rate below which redundancies built into the quantum computer can guarantee
accurate results with an arbitrarily high success rate. My work on continuous-variable
cluster states (described in later chapters) will not address fault tolerance directly because
it remains an open problem for that method of quantum computation. It is important
to be aware of the need for fault tolerance, however, since it is essential for any viable
implementation of quantum computation.
Resigning ourselves to the ubiquity of errors in any real-world situation, we can relax our requirement of exact universality to approximate universality. The finite gate
set {H, S, T, CX }, where the gates are defined above, is approximately universal for quantum computation, meaning that they generate a dense subset of SU(2N ). Operationally,
what’s important is that for any U ∈ SU(2N ), there exists a U ∈ SU(2N ) generated by
{H, S, T, CX } whose measurement statistics are the same as those of U when applied to an
arbitrary state |ψi, to within an arbitrarily small tolerance . In addition, these gates can
be implemented fault tolerantly.8
An important group of unitaries related to error correction and fault tolerance is the
Clifford group, which is just the normalizer of the Pauli group. In other words, the Clifford
group consists of all multi-qubit unitaries G that satisfy GP G† = P 0 , where P and P 0
are tensor products of Pauli operators: X, Y , Z, and/or 1 (the identity). Clearly, the
Pauli group is a subgroup of the Clifford group. The Clifford group can be generated
by the set {H, S, CX }. The Clifford group is closely related to quantum error correcting
codes [22, 36]. Furthermore, any quantum computation that uses only Clifford gates can
be efficiently simulated on a classical computer,9 a result known as the Gottesman-Knill
theorem [22, 36]. For universality, then, at least one single-qubit non-Clifford gate is needed
within the universal set. In the standard set, this is the π/8 gate T .
2.2.2
Quantum Circuits
A “quantum circuit” then is just like its classical counterpart: it is a functional composition
of elements of a universal gate set.10 In the quantum case, this is the structured application
of unitary gates to multiple qubits as input, which transform the input state of the qubits
8
Even though S = T 2 , and therefore the phase gate S could be left off of the list, it is included because
it is required for fault-tolerant implementation of the π/8 gate T . See Ref. [22] for more details.
9
This does not mean that quantum error correction can be performed by a classical computer, just that
the process can be simulated efficiently.
10
One important difference is that all quantum gates must be reversible (i.e., unitary). Strictly speaking
then, quantum circuits are the analog of classical circuits made up of reversible gates (the gates discussed
in Section 2.1 were not reversible).
15
into the desired output state, which can be measured in the computational basis to obtain
a classical answer if desired, or fed into another quantum circuit for further processing.
Like classical circuits, quantum circuits have a diagrammatic representation, an example of
which is given here:
|ψi
•
|0i
X
H
S
NM
m
(2.6)
Z m S |ψi
Key elements of this circuit are:
1. Input state |ψi ⊗ |0i, drawn on the left.
2. Quantum wires, drawn as horizontal lines and representing the “movement” of the
qubits “through the gates” (applying the associated unitary) as time passes, from left
to right.
3. Two-qubit controlled gate CX , drawn as a vertical line indicating the control qubit
(by a dot) and the gate X to be applied on the target qubit if the control is in the
state |1i.
4. Single-qubit gates, indicated by S and H in boxes, acting on the first qubit.
5. Projective measurement in the computational basis, indicated by the meter, and the
measurement outcome m ∈ {0, 1}.
6. Output state of the second qubit, Z m S |ψi, which depends on the measurement result m.
Clearly quantum circuits are reminiscent of their classical counterpart (example in Fig. 2.2).
More complicated circuits are possible, but Circuit (2.6) illustrates all the elements needed
to understand later material in this thesis.
In general, the minimum number of gates from a finite universal set required to (approximately) implement any given unitary U ∈ SU(2N ), which is a member of an N -indexed
family of such unitaries, is exponential in N , meaning that U cannot be implemented efficiently. This is true of most unitaries (using the above or any other universal gate set),
meaning that finding efficient quantum algorithms is a hard problem. In addition to this, in
order to be useful and worth the effort of building a quantum computer, the algorithm needs
to outperform its best known classical counterpart. Grover’s algorithm [13, 14] and Shor’s
algorithm [15] do this for unstructured database search and integer factoring, respectively,
but the number of “good quantum algorithms” is still quite small.
16
2.2.3
Implementation
So far, I haven’t talked at all about real-world implementation of quantum computing.
This is a huge area of experimental and theoretical interest, and many proposals have been
made for implementing qubit-based quantum computation [37]: polarized photons, nuclear
spins, trapped ions, liquid-state and solid-state nuclear magnetic resonance (NMR), neutral
atoms in optical lattices, quantum dots, spintronics, Josephson junctions, and even electrons
floating on liquid helium. None of these has emerged yet as a clear winner in the quest for
scalable quantum computing technology, but all proposals for implementing circuit-model
qubit-based quantum computation share certain essential characteristics:
• Qubits are used to represent, store, and process quantum information. Qubits are
physical quantum systems that can be modeled accurately as two-state systems. They
can be prepared in a fiducial initial state (usually |0i⊗N ). Quantum coherence can be
maintained within a single qubit, as well as between multiple qubits.
• Coherent unitary evolution can be implemented selectively for a single qubit, as
well as for multiple qubits together, in order to implement a universal gate set.
• Computational basis measurements allow for a classical answer to obtained at
the end of the computation through local projective measurements.11
In the next section, I will discuss replacing the need for coherent unitary evolution with
the requirement of a highly entangled resource state of many qubits called a cluster state
with which quantum gates can be implemented using only local projective measurements.
Following that, I will consider using continuonus-variable quantum systems instead of qubits
for quantum computation. The subsequent four chapters present my research on continuousvariable cluster states, which combines these two alternatives.
For more details about the standard circuit model of quantum computation, including
exact universality, fault tolerance, approximate universality, and quantum circuit diagrams,
please see Ref. [22]. More information on implementations can be found in Ref. [37]. At this
point, however, we will depart from this standard model to introduce two novel methods
of quantum computation that will provide the background for quantum computing using
continuous-variable cluster states.
2.3
One-Way Quantum Computation Using Cluster States
We now depart from the conventional model of quantum computation and introduce a
new model called one-way quantum computation. Invented in 2001 by Raussendorf and
11
Destructive measurements, such as photon counting, are allowed since included above is the requirement
that the qubits can be restored to a fiducial initial state.
17
Briegel [38], the model replaces the need for coherent unitary evolution—a major experimental obstacle, especially when implementing two-qubit gates—with a universal resource
state called a cluster state [39]. These states are highly entangled states of many qubits that
allow for local projective measurements alone to implement universal quantum dynamics.
To see how measurements alone can perform dynamics on quantum information, we
turn to the concept of teleportation. Ordinary quantum teleportation [40] uses an entangled
resource shared between two parties, along with local measurements made by one party (the
sender) to “teleport” an unknown quantum state from the sender to the the other party (the
receiver), up to a known unitary correction that is a function of the sender’s measurement
results. These results, being classical information, can be communicated over a classical
channel to the receiver, who can use them to apply a correcting unitary transformation and
restore the state that was sent. The details on this protocol can be found in Ref. [22], but I
wish to turn now to another, simpler type of teleportation that will be useful in illustrating
one-way quantum computation: one-qubit teleportation.
2.3.1
Quantum Wires
This description of one-way quantum computation follows closely the discussion in Ref. [41].
The one-qubit teleportation circuit looks like this [41, 42]:
|ψi
• H
|+i
•
NM
m
(2.7)
X m H |ψi
The strange-looking controlled operation (with apparently two controls and no target) is a
standard way of drawing the CZ operation in order to properly reflect its symmetry between
control and target qubits. The input state of the top qubit is an arbitrary state |ψi—even,
possibly, one that is unknown to the experimenter implementing this circuit. The input of
the bottom qubit is |+i =
√1 (|0i + |1i),
2
the +1-eigenstate of the X operator. The effect of
the CZ operation is to distribute the quantum information contained in |ψi to the pair of
qubits together. Measurement of the top qubit in the X-basis (measurement in the computational basis following a Hadamard H gate) now does not destroy this information but
instead leaves it in the bottom qubit, up to a known and measurement outcome-dependent
unitary modification.
It is impressive that the quantum information can be preserved in this distributed form
by the CZ operation and then recovered again by single-qubit measurement on the qubit
that used to contain the information. Also notice that the measurement result must be
known in order to get the state back, since the unitary modification (X m H) depends on
that result. While this is interesting, it’s not very useful yet. What we’d like is to be able to
perform dynamics on |ψi using just measurements. Ultimately, we’d like these operations
to include all possible single-qubit unitaries (or, at least, a universal set of such unitaries).
18
This is done by noticing that any unitary that is diagonal in the computational basis, which
we’ll call Zα , commutes with the CZ operations.
We know this circuit will perform a rotation by angle α in the XY-plane of the Bloch
sphere on the input state:
|ψi
Zα
NM
• H
m
X m HZα |ψi
•
|+i
(2.8)
Because a rotation by angle α in the XY-plane of the Bloch sphere has the form
Zα =
1
0
!
0 eiα
,
(2.9)
it commutes with the CZ gate, allowing Circuit (2.8) to be rewritten as
|ψi
• Zα
|+i
•
NM
H
m
(2.10)
X m HZα |ψi
All measurements on the top qubit following CZ can be implemented as a single measurement in the XY-plane of the Bloch sphere: α = 0 corresponds to measuring X; α = π/2
corresponds to measuring Y ; others correspond to measuring in another basis in the XYplane.
To apply any unitary to our input qubit, then, we only need to chain these circuits
together into measurements on a quantum wire. Using only single-qubit measurements on
this wire, we can perform arbitrary single-qubit dynamics on any input state |ψi:
|ψi
•
Zα
H
NM
m1
|+i
• •
Z±β
H
NM
m2
Z±γ
H
NM
m3
|+i
• •
|+i
•
(2.11)
Umod |ψi
where
Umod |ψi = X m3 HZ±γ X m2 HZ±β X m1 HZα |ψi
= (X m3 HX m2 HX m1 H) (Zγ HZβ HZα ) |ψi .
{z
}|
|
{z
}
known modification
19
desired unitary
(2.12)
Since the CZ gates commute, they can be applied in any order, and the “quantum wire”
is the state existing after the CZ gates but before the measurements. Measurement bases
are all in the XY-plane, chosen based on the desired unitary (which has Euler angles α,
β, and γ) and also on previous measurement results (the presence of ±). For instance,
the outcome of the first measurement m1 dictates whether the second measurement basis
for the following qubit is rotated either in the original direction (+β if m1 = 0) or the
opposite of it (−β if m1 = 1). This need for adaptiveness of measurement results is an
important feature of one-way quantum computation. A small amount of classical computing
is required to decide, based on previous results, which bases to measure later qubits in. The
end result, however, is that given the ability to apply arbitrary planar rotations, along with
a Hadamard in between each, any member of SU(2) may be applied to |ψi, up to a known
and measurement-dependent unitary modification [41].
The quantum wire discussed above has a graph representation, which is an important
concept in one-way quantum computation. Circuit (2.11) can be implemented using measurements alone on a cluster state with the following graph:12
@ABC
GFED
1
@ABC
GFED
2
@ABC
GFED
3
@ABC
GFED
4
(2.13)
where each node represents an initial |+i state, and links between nodes indicate that a
CZ gate was applied. The line to the left of node 1 indicates that this cluster state may
be attached to another like it in order to prepare the state |ψi at node 1. That is, since
Circuit (2.11) can be used to apply any unitary, another like it can be used to prepare
|ψi from |+i. Single-qubit, adaptive measurements on nodes 1, 2, and 3 leave Umod |ψi at
node 4.
2.3.2
Two-Qubit Gates
Even if longer chains are formed in this way, these quantum wires still only “carry” one
qubit of information. For universal quantum computation, a two-qubit gate like CX or CZ
is needed, requiring an interaction, or link, between two wires. Implementing CZ uses a
cluster state with an ‘H’-shaped graph, which includes a link between two quantum wires:
@ABC
GFED
1
@ABC
GFED
2
@ABC
GFED
3
@ABC
GFED
4
@ABC
GFED
5
@ABC
GFED
6
(2.14)
In fact, we can simplify this model by noticing that only the rightmost four nodes need to
be analyzed. Using the properties of quantum wires, we can propagate any state onto nodes
12
I follow the convention of Nielsen [41] and allow ‘cluster states’ to have any associated graph. In the
literature such states are sometimes called ‘graph states’, with a ‘cluster state’ restricted to one having a
square-lattice graph. I impose no such restriction, using the terms interchangeably.
20
2 and 5 through use of the techniques of Circuit (2.11) for each wire. Thus, our cluster for
performing a CZ gate can be considered simply as the following:
@ABC
GFED
2
@ABC
GFED
3
@ABC
GFED
5
@ABC
GFED
6
(2.15)
The lines to the left of nodes 2 and 5 remind us that a state will be teleported down two
quantum wires to arrive at nodes 2 and 5. When measurements are made on nodes 2 and 5,
the equivalent circuit is
node 2: |ψi
• • H
m2
NM
m5
(2.16)
•
3: |+i
5: |φi
NM
• • H
•
6: |+i
To determine the output of this circuit, we notice that the state after the CZ gate on nodes
2 and 5 is just CZ |ψi2 ⊗ |φi5 ⊗ |+i3 ⊗ |+i6 . Expanding the output of the CZ operation in
an arbitrary tensor-product basis,
CZ |ψi2 ⊗ |φi5 =
X
cij |ηi i2 ⊗ |ηj i5 ,
(2.17)
ij
we have remaining two one-qubit teleportations from node 2 → 3 and node 5 → 6. These
simply map |ηi i2 → X m2 H|ηi i3 and |ηi i5 → X m5 H|ηi i6 . Therefore, the output of this circuit
is just
(output) =
X
cij X m2 H|ηi i3 ⊗ X m5 H|ηj i6
ij
= X m2 H ⊗ X m5 H CZ |ψi3 ⊗ |φi6 .
(2.18)
We have effected a CZ gate on an arbitrary input state, followed by a known unitary
modification.
Universality in this model, then, follows from two things: (1) availability of quantum
wires, which can effect any single-qubit unitary, and (2) a graph that allows implementation
of two-qubit gates between quantum wires. Given these two requirements, adaptive singlequbit measurements alone can be used to perform universal quantum computation [38].
21
Two observations are in order at this point. First, only Clifford gates are required to
prepare a cluster state [41]. Using the Gottesman-Knill theorem [22, 36], this means that
if only Clifford gates are implemented using the cluster state, then the entire computation
can be simulated efficiently on a classical computer. Thus, we must be able to implement
at least one non-Clifford gate on our cluster state in order to use it for universal quantum
computation. The second observation is that the ubiquitous unitary correction imposed by
the one-way model at each step in the computation is always a Clifford-group correction.
As we shall see, this means that measurements implementing Clifford gates can be done in
parallel, since they commute with the measrement-dependent Clifford correction.
To see this, consider a quantum wire. Any operation that can be implemented using
the cluster state results in an output state of the form
(output) = X mN HZαN X mN −1 HZαN −1 · · · X m2 HZα2 X m1 HZα1 |ψi .
(2.19)
In general, the measurement bases, determined by αj , depend on the previous measurement
results in order to obtain the desired transformation on |ψi. If, however, α is multiple of π/2,
then given the fact that Znπ/2 = S n , we see that all the Zαj gates are Clifford gates. This
means that we can commute them through the other Clifford correction terms, resulting in
(output) = X mN HS nN X mN −1 HS nN −1 · · · X m2 HS n2 X m1 HS n1 |ψi
= P (mN , nN , . . . , m1 , n1 )HS nN HS nN −1 · · · HS n2 HS n1 |ψi ,
(2.20)
where P (. . . ) is now a Pauli correction that depends on the measurement results and on
the choice of measurement bases. The crucial point, though, is that the measurement bases
don’t depend on previous measurement results, since they are applied—interspersed with
H gates—directly to the input state |ψi, independent of the {mj }. This was not the case
for our implementation of a single-qubit unitary in Circuit (2.11), which used arbitrary (nonClifford) Z-rotations and required a ± correction depending on previous outcomes. This
result generalizes straightforwardly to two-qubit gates like CZ (described above) and CX
(described in Ref. [38]), which, along with H and S, generate the entire Clifford group. This
is a powerful result because it means that implementing Clifford gates requires no classical
feedforward of previous measurement results, and thus the measurements can be made in
any order, a property known as parallelism.
2.3.3
Universal Cluster States
It would be nice to know in advance whether a given graph satisfies the conditions for
universality. Recent work has addressed this issue thoroughly from a general standpoint [43],
but the original example [38] of a family of square lattice graphs (Fig. 2.3) is still the
canonical example of a universal family of graphs for one-way quantum computation. The
main property of these graphs that make them a universal family is that nodes can be
22
Figure 2.3: Square-lattice cluster state. Nodes represent qubits initially in the state |+i,
and links represent applied CZ gates between the linked qubits. This graph is a member of
a family of graphs—square lattices of arbitrary size—that is universal for one-way quantum
computation.
deleted from the graph simply by measuring in the computational basis. This changes the
coherent CZ gate into a Z m gate on all of the deleted node’s former neighbors in the graph,
where m ∈ {0, 1} is the measurement result. This is a local change of basis for the cluster
state, but up to this modification, the remaining qubits are still entangled in a cluster state,
but now with a modified graph due to the deletion. Using Z-measurements, undesired links
can be removed by deleting the qubits they connect to. In this way, quantum wires can be
“cut” from this graph, and links can be left between these wires for two-qubit gates.
Because every node is connected to each of its neighbors in a square lattice, however, it
is impossible to achieve an ‘H’-shaped graph (as in Diagram (2.14)) since there will always
be at least one additional node separating quantum wires cut from a square lattice. This
means that a CZ gate cannot be implemented as described when starting from a square
lattice. Nevertheless, the CX gate can be applied in this configuration, but with additional
technical details and a larger base graph. An explicit construction of the CX gate can
be found in the original proposal for one-way quantum computation [38], but it provides
no new fundamental insights beyond that for the CZ . In broad terms, a family of graphs
is universal for one-way quantum computation if its members allow for arbitrarily linked
quantum wires to be cut from them by deleting individual nodes with Z-measurements.
One-way quantum computation replaces the requirement of coherent unitary evolution
with possession of a cluster state on which local projective measurements can be made. The
conceptual departure from the circuit model lies in the replacement of physical qubits as
information carriers, which are manipulated with coherent unitary evolution, with “virtual
qubits” that get propagated through the cluster (and are simultaneously manipulated) via
measurements alone. The sequence of measurements to be made is determined by the
quantum algorithm to be implemented and the results of previous measurements in the
sequence.
The appeal of one-way quantum computation is that it consolidates most of the hard
work into creating the cluster state. Single-qubit projective measurements are often much
23
easier by comparison. In addition to this, some one-way quantum computation schemes
admit very high fault-tolerance thresholds [44, 45] (important for quantum computing in
the real world, as discussed in the previous section), and experimental realizations with
four qubits have already been achieved [46, 47]. Still, large-scale construction of such states
remains a difficult experimental problem. At this point, we leave qubits behind and turn
our attention to using continuous-variable systems for quantum computation.
2.4
Quantum Computation with Continuous Variables
While our ultimate goal is to replace qubits with continuous quantum variables in quantum
computation, it is helpful to generalize first to quantum computation with D-dimensional
systems called qudits.
2.4.1
Qudits
We begin this part of our journey with an observation: the Pauli group plays an important
role in qubit-based quantum computation—the computational basis consists of the eigenvectors of Z, and X flips between them. In generalizing to higher-dimensional systems, a
similar role will be played by the Weyl-Heisenberg group, which is the generalization of the
Pauli group.
We begin by defining an abstract “computational basis” for our qudits, which consists
of states labeled by the positive integers in base D: {|0iq , |1iq , . . . , |D − 1iq }. A subscript q
is used to distinguish this basis from the Fourier basis, which will be defined shortly. The
generalized Pauli (Weyl-Heisenberg) group is the group generated by the two shift operators13
X :=
X
|j + 1iqqhj|
and
Z :=
X
j
ω j |jiqqhj| ,
(2.21)
j
where ω := ei2π/D , addition in the X operator is modulo D, and |jiq is a computational
basis state for a single qudit. The X operator can be considered a cyclic shift in the
computational basis by one unit, so X k is a shift by k units. The computational basis is
the eigenbasis of Z, so application of Z applies a relative phase to each computational basis
state. Defining the Fourier transform operator
1 X jk
F := √
ω |jiqqhk| ,
D jk
13
The reason that Z is also called a “shift operator” will be demonstrated shortly.
24
(2.22)
we find that
F † ZF = X
F XF † = Z ,
⇐⇒
(2.23)
revealing F to be the qudit generalization of the H gate for qubits. Notice that unlike H,
the F gate is not Hermitian. If we further define the Fourier basis states
|jip := F |jiq ,
(2.24)
we can write X and Z in this basis as
X=
X
ω −j |jipphj|
and
Z=
j
X
|j + 1ipphj| ,
(2.25)
j
respectively. From this and from Eq. (2.21), it is obvious that while X cyclically shifts
computational basis states by one unit, Z cyclically shifts Fourier basis states by one unit.
Incidentally, since F = F F F † , the Fourier transform has the same form in the Fourier
basis. The use of the q and p subscripts allude to the eventual generalization in continuous
variables, where the computational basis states will be position eigenstates, and the Fourier
basis states will be momentum eigenstates.
Controlled operations generalize in an intuitive way to D-dimensional systems. The
controlled-X gate CX , where X is now given by Eq. (2.21), is defined as a controlled shift
among computational basis states in the target by an amount given by the computational
state of the control:
CX :=
X
|jiqqhj| ⊗ X j
j
=
X
|jiqqhj| ⊗ |k + jiqqhk| ,
(2.26)
jk
where the tensor product structure is (control⊗target). The controlled-Z gate CZ , where Z
is now given by Eq. (2.21), similarly becomes a controlled shift among Fourier basis states
in the target by an amount given by the computational state of the control:
CZ :=
X
|jiqqhj| ⊗ Z j
j
=
X
=
X
ω jk |jiqqhj| ⊗ |kiqqhk|
jk
Z k ⊗ |kiqqhk| .
(2.27)
k
This gate is invariant under exchange of the control and target qudits, just as in the qubit
case.
25
Universality for a qudit-based quantum computer is the ability to implement any member of SU(DN ), that is, any unitary on N qudits. In analogy with the qubit case, the
sets SU(D) ∪ {CX } and SU(D) ∪ {CZ } are universal, where CX and CZ have been generalized to their qudit definitions, Eqs. (2.26) and (2.27), respectively. More generally, all
single-qudit unitaries plus any imprimitive two-qudit operation forms a universal set, where
an imprimitive two-qudit unitary is one that does not map all product states to product
states [48]. Moreover, finite universal gate sets are also possible for qudit systems [49, 50].
An analogous result to the Gottesman-Knill theorem (see Section 2.2.1) holds for qudits,
as well: when the computation accesses only Clifford-group operations (those that normalize the Weyl-Heisenberg group), the entire computation can be efficiently simulated on a
classical computer [51].
2.4.2
Continuous Variables
From the D-dimensional case, the generalization to continuous variables is straightforward.
The discrete basis of states |jiq is generalized to a continuum of δ-function-normalized
eigenstates of the position operator q.Similarly the |jip states are generalized to δ-function
normalized eigenstates of the momentum operator p. The canonical commutation relation [q, p] = i (with ~ = 1) is assumed. Following Ref. [52] closely, we define the Fourier
transform operator in the CV case as
1
F := √
2π
Z
0 ds ds0 eiss s0 q qhs| .
(2.28)
In the CV case, the weyl-Heisenberg group is a Lie group. We define the analogs of X
and Z through their actions on position- and momentum-basis states, respectively. In
the discrete case, X j shifts a position eigenstate cyclically by j units, and Z j similarly
performs a momentum shift. Since p generates shifts in position, and −q generates shifts
in momentum, we will define the following position- and momentum-shift operators:
X(s) := e−isp
and
Z(t) := eitq ,
(2.29)
respectively. Now we have X(s)|s0 iq = |s0 + siq , and Z(t)|t0 ip = |t0 + tip . Note the crucial
difference from the qudit case: these shifts can be infinitesimal, X(δs) ' 1−iδs p and Z(δt) '
1 + iδt q, respectively. The relation between these operators under Fourier transform is
exactly analogous to the D-dimensional case, too:
F † Z(s)F = X(s)
⇐⇒
F X(t)F † = Z(t) .
(2.30)
The form of the controlled-X and controlled-Z gates can be deduced intuitively by considering their action on appropriate basis states. The controlled-X operator performs a
26
position-shift on the target system by an amount dependent on the eigenvalue of a positionmeasurement on the control system. The generator of this transformation is q ⊗ p, since
exp(−iq ⊗ p)|s1 iq ⊗ |s2 iq
= |s1 iq ⊗ exp(−is1 p)|s2 iq
= |s1 iq ⊗ |s2 + s1 iq ,
(2.31)
which is just what we want. A similar calculation reveals that −q ⊗ q generates the
controlled-Z operation. Thus, we have
CX := exp(−iq ⊗ p)
and
CZ := exp(iq ⊗ q) .
(2.32)
Again, the symmetry with respect to the two systems is evident in the CZ gate.
Universality for continuous-variable quantum computation is, as one would expect, the
ability to implement any unitary operation on N continuous-variable systems. Also as
expected, all single-system unitaries plus either CX or CZ from Eq. (2.32) is exactly universal [52].
The analog of the Clifford group in the continuous-variable case is the set of Gaussian
operations. This unitary group is the normalizer of the Weyl-Heisenberg group, and the
corresponding Lie algebra consists of all homogeneous quadratics in q and p. By conjugation, Gaussian operations act linearly on the Weyl-Heisenberg algebra, which consists of
all linear combinations of q and p and thus transform phase-space displacements into other
displacements. Therefore, this group is often referred to as the group of linear transformations. The operations are called ‘Gaussian’ because they map all states with Gaussian
Wigner functions to other states with Gaussian Wigner functions. Gaussians include the
Weyl-Heisenberg group (phase-space displacements), rotations and shears (“squeezes”) in
phase space, as well as controlled gates like CX and CZ .
Being the continuous-variable analog of the Clifford group, Gaussian operations satisfy
a generalization of the Gottesman-Knill theorem: any continuous-variable quantum computation that uses only Gaussian operations (and initial Gaussian states) can be efficiently
simulated on a classical computer [53]. This means that Gaussians alone are not universal
for quantum computation. Universality requires that at least one non-Gaussian unitary be
present in the continuous-variable gate set.
Canonical examples of continuous-variable quantum systems are quantum mechanical
oscillators and quantum fields, especially the electromagnetic field. In this case, Gaussian
operations correspond exactly to those of linear optics—phase shifters, beam splitters, and
squeezers—so called because they correspond to linear transformations on phase space.14
14
Appendix A contains a very brief reference for some of the key quantum optics concepts used in this
thesis, but it is not sufficient as an introduction for the novice. Such an introduction may instead be found
in numerous textbooks on the subject, e.g., Ref. [54].
27
The generalized Gottesman-Knill theorem implies, though, that some nonlinear optical
element (e.g., single-photon states, photon number-resolving measurements, etc.) must be
available for optical universal quantum computation.15
Fault tolerance remains an open problem for continuous-variable quantum computation. Some studies have addressed continuous-variable error correction [56, 57, 58, 59, 60],
but a finite fault-tolerant threshold is not known to exist. Given the ubiquity of experimental errors, the bandwidth of any continuous-variable quantum computer is most likely
finite, which motivates the study of encoding a lower-dimensional system (e.g., a qubit) into
each oscillator or field mode [57, 61]. This would allow for “continuous-variable quantum
hardware” to be used for standard qubit algorithms and suggests the possibly that fault
tolerance could be addressed within the qubit framework [62]. Nevertheless, fault tolerance
more generally in the continuous-variable case remains an important open question.
2.4.3
Continuous-Variable Cluster States
The next four chapters describe a model of quantum computation that combines the methods described in this and the previous sections. The model is one-way quantum computation using continuous-variable systems. The universal resource is a continuous-variable
cluster state. Adaptive single-mode projective measurements allow for universal quantum
computation using continuous variables. Chapter 3 follows closely Ref. [1], which is the
initial proposal for continuous-variable one-way quantum computation and includes a proposed optical implementation. Chapter 4 describes a mathematical result allowing for
continuous-variable cluster states to be generated in an extremely compact experimental
setup consisting of a single optical parametric oscillator (OPO) [2]. While this demonstrates
that, in principle, one can always make useful cluster states this way, it says nothing about
the optimal method of doing so, leaving open the possibility that the method may still suffer
from poor scaling due to the need for complicated OPO design. Chapter 5 demonstrates the
power of this method, illustrating that a very simple and scalable OPO design will allow for
generation of many copies of small continuous-variable cluster states [3]. While the family
of states generated in this manner is not universal for quantum computation (because only
the number of copies of the graph grows, not the graph size), this result nevertheless justifies
enthusiasm for this method of construction. Chapter 6 is the clincher, which shows how to
construct a cluster state that is universal for continuous-variable quantum computation in
an extremely scalable fashion from a single OPO [4]. We turn now to one-way quantum
computation with continuous-variable cluster states.
15
This may seem to contradict the existence of “linear optical quantum computing,” introduced by Knill,
Laflamme, and Milburn (KLM) [55]. This is a qubit scheme, so I won’t discuss it more than to say that
the nonlinear elements involved are single-photon sources and photon counting measurements, while all
intervening optical elements are linear.
28
Chapter 3
Universal Quantum Computation
with Continuous-Variable Cluster
States
3.1
Introduction
Unifying the concepts of one-way quantum computation (Section 2.3) with continuousvariable quantum computation (Section 2.4), in this chapter, I describe a model of universal
quantum computation (QC) using continuous-variable (CV) cluster states, which closely
follows the work published in Ref. [1] by myself and coauthors. Included is a proposal for
an optical implementation of this scheme where squeezed light sources serve as the nodes
of the cluster. The main advantage of this approach is that not only can computations
with the cluster be performed deterministically, but also the preparation of the cluster
state, including connecting the nodes, can be done unconditionally. This is in contrast
to the discrete-variable linear-optics schemes [63, 64, 65], where cluster states are created
probabilistically. Therefore, the CV approach appears to be particularly suited for further
experimental demonstration of the general principles of cluster-state QC.
In the optical implementation, once the cluster state has been created, single-mode
homodyne detection alone will allow for any multi-mode Gaussian transformation to be
performed on the information contained within the cluster. Analogously to the implementation of Clifford gates using qubit clusters, the homodyne detections can be done in any
order, a property known as parallelism. For universal QC, in addition, only one single-mode
non-Gaussian projective measurement (e.g., photon counting) is required. However, parallelism no longer applies to non-Gaussian measurements, because the choice of subsequent
measurement bases will depend on the outcome of earlier measurements. This adaptiveness
of the measurement bases is again analogous to the qubit case when computing non-Clifford
gates. While CV cluster states have been described previously in [66], it is claimed there that
29
such states are an insufficient resource for universal QC because of their Gaussian character [67]. In fact, they are sufficient as long as we can perform a non-Gaussian measurement.
An analogous result holds for qubit cluster states, which can be created entirely using Clifford group operations [51] but are nevertheless universal once a non-Clifford measurement
is allowed.
Although CV cluster states can be built deterministically, it will be impossible to create
perfect CV cluster states due to the finite degree of squeezing obtainable in the laboratory.
This results in distortions to the quantum information as it propagates through the cluster
state. These distortions (along with other errors) are discussed. An experiment is proposed
to demonstrate how parallelism and post-selection can be used to mitigate these effects
when implementing Gaussian operations. Since the entire field of continuous-variable oneway quantum computation began with the results presented in this chapter [1], there remain
many open questions and unexplored research avenues. A look at the research landscape is
included at the end of the chapter.
3.2
Continuous-Variable Cluster States
Other authors [68, 69, 70] have extended the cluster-state formalism to d-level systems (qudits). Here we generalize these results to CVs, whose use in QC was discussed in Section 2.4.
The essence of the qubit cluster-state model of QC lies in the one-qubit teleportation circuit,
Circuit (2.7) in Section 2.3. This circuit gives the ability to teleport operations diagonal
in the computational basis onto the state in question after the cluster has been prepared.
This allows dynamics to be performed solely through measurement. The CV analog of the
one-qubit teleportation circuit is
|ψi
• D
F†
NM
s
(3.1)
|0ip
•
X(s)F D |ψi
R
In this diagram, |0ip = (2π)−1/2 ds|siq is a zero-momentum eigenstate (the generalization
of |+i), the controlled operation indicated is a CZ gate, Eq. (2.32), and D is any operator
diagonal in the computational basis (i.e., of the form exp[if (q)]). The projective measurement is of q and yields a real number s, which becomes the argument of the displacement
X(·) at the output of the circuit. The essential feature of this circuit is that the CZ gate
commutes with any diagonal operator D. This means that even though D is applied after the CZ gate, it acts as if it had been applied before. Since the operations D and F †
followed by computational basis measurement are equivalent to a single measurement of
D† pD, manipulating quantum information in the CV cluster is possible through projective
measurements alone. Concatenation of these circuits makes it possible to implement any
single-mode unitary [52].
30
As is the case for qubits [71], every CV cluster state has a graph state representation,
where each node in the graph is a separate CV mode, and each link in the graph represents
a CZ that has been performed between the corresponding nodes (systems). Linear graphs,
where each node has at most two links, can be used for single-mode evolutions, but not
multi-mode gates. The simplest implementation of a CZ gate involves a graph state with a
link between two adjacent quantum wires:
@ABC
GFED
1
@ABC
GFED
3
@ABC
GFED
2
@ABC
GFED
4
(3.2)
The lines to the left of nodes 1 and 2 indicate that a bipartite state |ψi will be teleported
down two quantum wires to arrive at nodes 1 and 2. In analogy to the calculation in
Section 2.3.2, measuring p on nodes 1 and 2 leaves
(X(s1 )F ⊗ X(s2 )F )CZ |ψi
(3.3)
on nodes 3 and 4.
A small set of Hamiltonians that are polynomials in q (e.g., {q, q 2 /2}), along with the
Fourier transform, are sufficient to implement any single-mode Gaussian [52]. Furthermore,
adding the ability to perform a CZ operation (as described above) allows implementation of
all multi-mode Gaussians. While this is not sufficient for universal QC, given an encoding
that maps all qubit Clifford operations to CV Gaussian operations (the GKP encoding being
one example [57]), this would be sufficient for many quantum error correction protocols [36].
Adding to the toolbox any single non-Gaussian projective measurement allows for universal
QC using CV cluster states [52].
3.3
Optical Implementation
Since each plane-wave mode of the electromagnetic field behaves as an independent harmonic oscillator, we can use these modes as CV systems for our CV cluster state (see
Appendix A). To do this, we choose eigenstates of the quadrature-amplitude operator q ∝
(a + a† ) to be the computational basis.1 The commutation relations
[a, a† ] = 1
and
1
[q, p] = i ,
(3.4)
Quantum optics quadratures are defined only up to an overall phase. Once a phase convention for a
given mode is chosen, however, it must be used consistently throughout.
31
with ~ = 1, are satisfied by the definitions
1
q = √ (a + a† )
2
−i
p = √ (a − a† )
2
and
(3.5)
for each mode, where the operators are implicitly assumed to be in the interaction picture so
that freely propagating states of the field remain fixed with respect to them. In this unitless
convention, the variance of the vacuum state (which can be measured experimentally using
homodyne detection) is given by hq 2 i = hp2 i = 1/2.
Construction of an ideal CV cluster state requires zero-momentum eigenstates, which
cannot be normalized and are thus unphysical. In this optical model, they represent infinitely squeezed vacuum states, which require infinite energy. We can approximate them,
though, by finitely squeezed vacuum states:
2 −1/4
Z
|0, Ωip := (πΩ )
2 /2Ω2
dt e−t
|tip ,
(3.6)
with Ω2 < 1 being the variance of a Gaussian wave packet in momentum space (with
hp2 i = Ω2 /2). The states |0, Ωiq are defined analogously with p → q in Eq. (3.6). Note
that |0, Ωi = 0, Ω−1 . The fact that these states are finitely squeezed means that we
p
q
will not have perfect fidelity while propagating quantum information through our cluster.
This will be addressed later. Given the graph state that we wish to create, we need one
independently squeezed mode per node, and we need the ability to perform a CZ gate
between modes in accordance with the graph. This operation is a quantum nondemolition
(QND) interaction [54] and can be implemented using two beamsplitters and two in-line
squeezers [72]. Alternatively, it could be directly realized via a linearized optical-fiber crossKerr interaction [73]. (See also Sec. III of Ref. [66] for further ideas.)
Propagation down a quantum wire (D = I) is achieved through momentum-quadrature
homodyne detection. As discussed previously, multi-mode Gaussian operations require only
that we can apply D = eisq and D = eitq
2 /2
for all s, t ∈ R. Applying a gate D to the
encoded state is achieved by measuring the operator D† pD. Thus, the Z(s) = eisq gate is
implemented by measuring the operators
Z(−s)pZ(s) = p − s .
(3.7)
This is trivial to implement: simply measure p and subtract s from the result. The gate
denoted P (t) = exp(itq 2 /2) is implemented by measuring
P (−t)pP (t) = p + tq .
32
(3.8)
Notice, however, that by defining θ = tan−1 (−t), we can rewrite this operator as
P (−t)pP (t) =
p cos θ − q sin θ
,
cos θ
(3.9)
which is simply homodyne detection in a rotated quadrature basis, followed by a rescaling
of the measurement results by a factor of cos θ = (1 + t2 )−1/2 . Thus, once the cluster has
been prepared, we are able to perform all multi-mode Gaussian operations simply through
homodyne detection.
Furthermore, analogously to implementing Clifford group operations on qubit cluster
states, all multi-mode Gaussian operations may be implemented on CV clusters with the
appropriate measurements made in any order. Performing the measurements in a different
order is equivalent to commuting Gaussian operations through the (Gaussian) measurementdependent corrections, resulting in different corrections, but leaving the measurement bases
unchanged. This is known as parallelism in cluster-state QC [74]. Non-Gaussian operations
in general cannot be parallelized, since later measurement bases will depend on current
measurement results, a property known as adaptiveness.
Universal QC requires the ability to implement at least one non-Gaussian operation [52].
In our case, this will be achieved through a measurement in a non-Gaussian basis. While
one can, in principle, use the continuous degree of freedom directly for QC, it will almost
certainly be more practical (considering experimental errors) to encode finite dimensional
systems in the CV modes, e.g., as in the GKP proposal [57], which encodes one qubit into
each oscillator. In this case, the optimal non-Gaussian operation would be tailored to implement a desirable non-Clifford unitary in the qubit space. Photon counting is one possibility
and fits nicely into the cluster formalism since it is already a projective measurement. Another option is to measure in a nonlinear polynomial basis, such as that corresponding to
the observable p + uq 2 for any one particular choice of u. This is equivalent, in the language
of Circuit (3.1), to implementing the gate D = eiuq
3 /3
. The GKP proposal discusses both
options in more detail. At this point, the questions of encoding scheme and non-Gaussian
measurement are left to future work.
3.4
Errors Due to Finite Squeezing
Possible sources of experimental error include the finite squeezing of the input states, mixed
input states (but still Gaussian), and distortions due to the QND operation used to form
the cluster. Since any physical implementation of our protocol will be forced to use finitely
squeezed states (because of finite energy requirements), we will consider the effects of finite
squeezing in some detail.
Finite squeezing in Eq. (3.6) modifies the output of the circuit in Circuit (3.1) to
MX(s)F D |ψi, where M is a distortion that applies a Gaussian envelope in position space
33
with zero mean and variance Ω−2 :
Z
M |ψi ∝
ds es
2 Ω2 /2
|siqqhs|ψi .
(3.10)
Notice that this is not a unitary transformation, and the state must be renormalized after
this envelope is applied. This is also equivalent to convolution in momentum space by a
Gaussian with variance Ω2 . Mixed input states can be accommodated in this analysis (in
the Wigner representation) simply by allowing the convolution width to be independent
of the width of the Gaussian envelope. Thus, the transformation implemented by each
measurement, which used to consist solely of F s, Ds, and phase-space displacements, now
includes a ubiquitous distortion at each step in the evolution. The severity of this distortion
depends inversely on the amount by which the sources are squeezed.
Concatenated circuits of the form (3.1) apply the transformation
· · · MX(s2 )F D2 MX(s1 )F D1 |ψi
(3.11)
to the input. Alternatively, we can gather the fixed distortions to one end of the operation
and transform this into the useful form
f 1 , . . . , sn ) |ψi ,
U0 (s1 , . . . , sn )M(s
(3.12)
where U0 is the unitary that would be applied in the case of an ideal cluster, and
0
f 1 , . . . , sn ) |ψi
ψ ∝ M(s
(3.13)
is a distorted initial state, with the distortion now depending on both the measurement
results and the gate to be implemented. The effect is the same for multi-mode gates: at
each measurement step, a fixed distortion M is applied to each mode. Specifically, in the
case of the CZ gate, the resulting output is
MX(s1 )F ⊗ MX(s2 )F CZ |ψi .
(3.14)
The distortion operations in the multi-mode case can similarly be gathered to the right
while becoming measurement- and gate-dependent. Errors in the QND operation can be
modeled as additive Gaussian noise, which has a similar distorting effect, the strength of
which scales as the number of links in the cluster’s graph.
3.5
Experimental Proposal for Cluster-Based Error Reduction
34
Parallelism, which is a feature particular to cluster-state QC, along with post-selection, can
be used to reduce the impact of the errors described in the last section when implementing
Gaussian operations. The following experiment is proposed to demonstrate this. For concreteness, consider a linear cluster of five nodes (although any number greater than three
will suffice):
@ABC
GFED
1
@ABC
GFED
2
@ABC
GFED
3
@ABC
GFED
4
@ABC
GFED
5
(3.15)
(The line to the left of the first node indicates where this cluster might be attached to
another one.) Many simple Gaussian operations may be implemented on this cluster through
homodyne detection on the first four nodes (see Section 3.3). With each such measurement
there is the possibility of the resulting distortion severely affecting the quantum state in
a measurement-dependent way (see the previous section). However, if we choose to delay
applying the QND operation (CZ ) that connects node 1 to node 2, we can isolate nodes
2–5 into a “mini-cluster,” which is separate from the quantum state to be acted upon.
By measuring nodes 3 and 4 before attaching the mini-cluster to the input state, we can
calculate the effect of the distortion from these two nodes before that distortion ever affects
the state. If this distortion does not preserve the Wigner phase-space region likely to be
occupied by the input state (which depends on the chosen encoding), we discard this minicluster and try again. If it does, we perform the QND operation to attach nodes 1 and 2.
We now have only two “dangerous” measurements to make (on the newly attached nodes)
instead of four, with the output appearing on node 5. State tomography can be used to
compare the fidelity of these two approaches.
This technique generalizes easily to multi-qubit operations and can, in fact, be applied
to mini-clusters implementing any Gaussian operation. The greatest benefit will be for
Gaussians that require many measurements. While this proposal has “bent the rules” of
cluster-state QC a bit by delaying attachment of the mini-cluster and by post-selecting miniclusters based on measurement results, this may yet prove to be a practical procedure for
dealing with experimental errors. This result has the flavor of Ref. [44], wherein it is shown
that through post-selection the reliability of an error-correcting ancilla cluster (called a
“telecorrector”) can be guaranteed before it is attached to the state to be corrected. Finally,
it is worth noting that this protocol will most likely require a non-Gaussian encoding of
qubits, although an in-principle demonstration could be made with Gaussian inputs.
3.6
Conclusion
This chapter discusses the generalization of one-way quantum computation to continuousvariable systems, including a proposed optical implementation that uses squeezed light
sources and quantum nondemolition operations to build a Gaussian cluster state. Homodyne detection alone suffices to implement all multi-mode Gaussian operations using
35
the cluster state, with the addition of one non-Gaussian measurement allowing for universal quantum computation. Many of the properties of qubit-cluster computation also
apply to the continuous-variable case, including parallelism and adaptiveness. Within the
continuous-variable approach, a lower-dimensional encoding scheme will most likely be required for experimental viability. Due to their Gaussian nature and deterministic method
of construction, we expect that continuous-variable cluster states will allow for further experimental demonstrations of the principles of cluster-state quantum computation. Such an
experiment was proposed to demonstrate improvement in the fidelity of Gaussian operations
using post-selection and parallelism.
This work opens the door to an entirely new model of quantum computation. As
such, there remain numerous avenues for research stemming from it. An obvious extension
is to study particular methods of implementing any given operation using a continuousvariable cluster state. This has been achieved for single-mode Gaussian transformations
using a quantum wire [75], but the particular form of more general transformations is still
needed. The issue of the nonlinear measurement is an important question if this method
is to be useful as a viable implementation for quantum computation. As discussed above,
this will likely be tied to the qubit (or other) encoding chosen and will likely depend on
the availability of fault-tolerant implementations. The GKP encoding has some amount of
fault tolerance built in [57, 59], but it is not yet clear whether this will help reduce errors
due to imperfect cluster states or whether such states can every be made in the laboratory.
Another possibility is a coherent-state encoding of qubits [61, 62], which is easier to use
experimentally but whose fault-tolerance properties are also unknown in this context.
Implementation has come a long way since this original proposal, with simplifications
that eliminate the (difficult) QND interactions [76] in favor of just beamsplitters [77] and
my work, with experimental colleagues, on generating them from a single optical parametric
oscillator (OPO) [2, 3, 4], described in the next three chapters. In light of these developments that eliminate the use of QND interactions altogether, the error reduction scheme
described in Sec. 3.5 would need to be revised or replaced. Another concept that arises from
my work on single-OPO implementation is the notion of a continuous-variable cluster state
with a weighted graph, where the weights indicate different correlation strengths between
different node pairs. While it is known that weighted graphs are universal in the ideal case,
the effect of imperfect clusters must be generalized to this case.
Clearly, there is much to be done in this area of research. Over the next three chapters,
we travel steadily in one particular direction, demonstrating the feasibility of generating
large continuous-variable cluster states in an experimentally compact manner.
36
Chapter 4
Ultracompact Generation of
Continuous-Variable Cluster States
4.1
Introduction
The goal of this chapter, which describes the work reported in Ref. [2], is to improve on
the originally proposed method of constructing continuous-variable cluster states, which requires single-mode squeezers and (experimentally arduous) quantum nondemolition (QND)
interactions (Chapter 3). The improvement described in Ref. [77], requiring only singlemode vacuum squeezers (much easier) and beamsplitters is also surpassed by this proposal,
which uses only a single optical parametric oscillator (OPO) that performs all the required
squeezing interactions at once. While not obviously advantageous due to the possible need
for a complicated OPO, the main result is used in Chapters 5 and 6 to demonstrate the
power of the method for scalable creation of continuous-variable cluster states.
The proposal for constructing continuous-variable cluster states, as described in Chapter 3, can be summarized to the following. First, prepare each mode (represented by a
graph vertex) in a phase-squeezed state, approximating a zero-phase eigenstate (analog
of Pauli-X eigenstates). Then, apply a quantum nondemolition (QND) interaction of the
form exp(iqj qk ) (analog of controlled-phase CZ ) to each pair of modes (j, k) linked by
an edge in the graph. All QND gates commute (as do CZ gates), so the full multimode
QND operator to be applied is exp( 2i qT Aq), where q = (q1 , . . . , qN )T is a vector of amplitude quadrature operators, and A is the adjacency matrix for the graph.1 The resulting
continuous-variable (CV) cluster state is a member of a family of squeezed states indexed by
an overall squeezing parameter r > 0 for which the variance of each component of (p − Aq)
tends to 0 as r → ∞ [77]. Analogous to the definition for q above, p = (p1 , . . . , pN )T is
1
An adjacency matrix, associated with a given graph, is a matrix whose rows and columns are labeled by
nodes in the graph and whose nonzero entries Ajk describe a link between nodes j and k with weight Ajk ,
which is usually taken to be a real number (although more abstract weights are conceivable; matrix-valued
weights are used in Chapter 6). This thesis is concerned with undirected graphs only, which have symmetric
adjacency matrices (A = AT ). An unweighted graph has all weights equal to 1.
37
a vector of phase quadrature operators. We’ll write the defining relation for a CV cluster
state family as
p − Aq → 0 .
(4.1)
The CV cluster states described in Chapter 3 satisfy this relation, but they are not the only
ones that do. Other Gaussian states exist that are strongly squeezed along the indicated
quadratures but are nevertheless not equivalent to those constructed by the method described above [77]. We shall also consider these states to be valid CV cluster states, which
is consistent with the definition used in Ref. [77]. Recall that while only Gaussian operations are needed to create CV cluster states from the vacuum, using them for universal QC
requires that at least one single-mode non-Gaussian measurement (such as photon-number
resolving detection) be available (Chapter 3).
Although convenient theoretically, the above procedure is not optimal for experimental implementation because the QND gates contain in-line squeezers (seeded OPOs) [76],
which are difficult to implement. It can be spectacularly simplified [77] by use of the BlochMessiah reduction [72], which transforms any Gaussian operation into the canonical form of
a set of single-mode squeezers sandwiched between two multimode interferometers. With a
vacuum input, the initial interferometer is irrelevant and any Gaussian CV N -mode cluster
can be formed by N single-mode vacuum squeezers (easier to implement than in-line ones)
followed by a network of O(N 2 ) beam splitters—i.e., a quadratically large, stable interferometer [77]. Recently, a four-mode linear cluster state was demonstrated experimentally
using this method [78, 79].
This chapter shows that it is, in fact, possible to integrate all single-mode OPOs into
one multimode OPO, pumped by an O(N 2 )-mode field, and to eliminate the beam splitter
network completely. This is equally resource-efficient as the proposal in Ref. [77] but the
complexity has been shifted from a stabilized O(N 2 )-element interferometer (unwieldy for
large N ) to the nonlinear medium of a single OPO and the frequency content of the pump
beam. This scheme is much more compact, and it has interesting prospects for scalability because it effectively represents the quantum entangled version of an optical frequency
comb: as is well known, a femtosecond laser effectively compactifies ∼ 105 phase-locked
continuous-wave lasers into a single beam [80, 81]. It has been shown that such a comb can
R
be transformed into a GHZ state dx |xi1 · · · |xiN , where the subscripts denote consecutive comb lines, using a complete network of concurrent nonlinear interactions [82, 83, 84],
and the nonlinear medium required to create four-mode entanglement in a single OPO has
already been demonstrated [85]. Engineering concurrent nonlinear interactions between an
arbitrary number of modes is a complicated problem but it is now solvable in the general case by use of generalized quasi-phase-matching in photonic quasicrystals [86]. This
enables arbitrarily difficult nonlinear interactions (e.g., simultaneous generation of the sec-
38
ond, third, and fourth optical harmonics, all in different directions [86]) to be engineered in
a single OPO.
The central result of this chapter is a mathematical connection between CV cluster states
and two-mode squeezing (TMS) graph states [84]. CV cluster-state graphs have vertices
representing phase-squeezed states and edges corresponding to QND unitary interactions
and are the ones we wish to implement for one-way QC. TMS graphs have vertices representing vacuum inputs and edges denoting individual terms in the multimode squeezing
Hamiltonian [82]
H = i~κ
X
Gmn (a†m a†n − am an ),
(4.2)
m,n
where G denotes the adjacency matrix of the TMS graph, κ is an overall nonlinear coupling
constant (squeezing parameter per unit time), and we have indicated ~ explicitly. We prove
that the two are related: any CV cluster state with a bicolorable graph can be created by
applying a single multimode squeezing Hamiltonian of the form of Eq. (4.2) and any such
Hamiltonian generates some CV cluster state. We detail how to create a square cluster
using this method with current technology.
Before going on, we should note that while we have made reference to an optical cavity
and used that picture to restrict the modes in Eq. (4.2) to a discrete set, we have not
yet explicitly included the cavity damping terms (see Appendix A). This has important
implications for above-threshold operation of the OPO [83] but is less troublesome for
operation below threshold [82, 85] in the current context. Still, it is well-known that an
OPO generates bipartite entanglement both below [87, 88] and above [89, 90, 91] the OPO
threshold; our goal is to extend this to graph-like entanglement structures. Nevertheless, it
will be important—once we have shown the connection here and then later expanded it to
a meaningful proposal for large-scale CV cluster state generation (Chapters 5 and 6)—to
properly design an experimental setup to couple the entanglement generated within the
cavity to the external modes. The first (and essential) step, of course, is to engineer an
appropriate Hamiltonian interaction to be facilitated by the nonlinear medium; hence our
focus on the Hamiltonian in Eq. (4.2).
4.2
CV Clusters and Multi-Mode Squeezing Hamiltonians
Given a target CV cluster state, our goal is to effect a transformation on the quadrature
operators such that Eq. (4.1) holds for the new quadratures. We first collect q and p into
a column vector x = (q1 , . . . , qN , p1 , . . . , pN )T . Gaussian transformations on the vacuum in
Hilbert space correspond to symplectic linear transformations on this vector in the Heisenberg picture [92, 93]. We denote by Uα the symplectic transformation corresponding to a
unitary that creates a CV cluster state from the vacuum. The level of overall squeezing is
39
represented by α > 0, which should be as large as possible. From Eq. (4.1) we have
−A 1 Uα x0 → 0 ,
(4.3)
where the block matrix above is N × 2N , Uα is 2N × 2N , and x0 is the vector of quadrature
operators representing the vacuum state. The arrow denotes the limit α → ∞.
We have some additional freedom in Eq. (4.3). After the transformation Uα is applied,
we can perform arbitrary phase shifts for each individual mode at the output, which we
will represent with the matrix T. This is a passive transformation on the state, which
can be effected simply by reinterpreting the output modes (i.e., no change to the physical
apparatus used to create the state is required). Therefore, we have that
−A 1 TUα → 0
(4.4)
is sufficient to conclude that Uα can be used to create a CV cluster state with adjacency
matrix A from the vacuum.
As we will now show, if A represents a bicolorable graph, we can always do this with
a multimode squeezing Hamiltonian. By definition, the nodes of a bicolorable graph are
partitioned into two sets such that all graph edges link one set to the other. These graphs
are called bicolorable because the two sets (and the nodes each contains) can be assigned
different colors. Bicolorable graphs include the square lattice graph of arbitrary size, which
is universal for QC, and any of its subgraphs. Star graphs (of any size) are also bicolorable,
with the node at the center being one color and the rest a different color. As a counterexample, the triangle graph (and, more generally, any graph with an odd cycle in it) is not
bicolorable.
Consider a multimode squeezing Hamiltonian given by Eq. (4.2), where G is the (as
yet, undefined) adjacency matrix for a TMS graph. Writing Uα as the Heisenberg matrix
corresponding to exp(−iαH) gives
Uα =
eαG
0
0
e−αG
!
.
(4.5)
A large (but finite) value of α is required for a useful CV cluster state. Although previous
work [82, 83, 84] has emphasized uniformly weighted TMS graphs with no self-loops, at this
point the only restriction we are going to place on G is that it be symmetric and full-rank.
Experimental requirements will favor some G’s over others but, since any G is in principle
possible to implement [86], we will not impose any additional restrictions at this point.
With these requirements we can write G as the difference of two positive semidefinite
matrices that are mutually orthogonal. By this we mean G = G+ − G− , where G± ≥ 0
and G± G∓ = G∓ G± = 0. We write G◦± for the Moore-Penrose pseudoinverse of G± , which
(for symmetric matrices) is obtained by inverting all the nonzero eigenvalues of G± . Then,
40
G−1 = G◦+ − G◦− . The projectors onto the positive and negative subspaces of G are P± =
G± G◦± = G◦± G± . Recalling Eq. (4.5), we need both the positive and negative exponentials of
G in the limit of large α. In the positive (negative) case, such an operation will magnify all
the positive (negative) eigenvalues of G and zero out all of G’s negative (positive) eigenvalues.
To write this concisely, we start with the fact that e−αG± → P∓ for large α, since all of the
nonzero eigenvalues of G± get sent to zero (since G± ≥ 0) while the zero eigenvalues get
raised to 1. This gives
e±αG = e−αG∓ eαG± → P± eαG± = G± G◦± eαG± .
(4.6)
By suitably numbering nodes, the adjacency matrix for any bicolorable graph can be written
as
A=
0
A0
AT0
0
!
,
(4.7)
where A0 is L × (N − L). Instead of using colors, we will label the first L modes by +
and the rest by − because the number of each will correspond to the number of positive
and negative eigenvalues of G, respectively. Recalling Eq. (4.4), we will use the phase-shift
freedom in T to rotate all of the − modes by −π/2 and leave the others unchanged. This
gives
−A0 1 0
0
−A 1 T =
−AT0
0
=
−AT0
!
0 1
!
0 1 A0
1−
0
1 0
0
1+ −1−
!
1+
,
(4.8)
where 1± is the identity matrix on the ± modes and zero on ∓ modes and the identity
blocks and zero blocks are sized appropriately, according to the dimensions of A0 . Plugging
Eqs. (4.5), (4.6), and (4.8) into Eq. (4.4) gives the following sufficient condition for cluster
state creation:
0
−AT0
0 1 A0
1 0
!
0
G+ G◦+ eαG+
0
0
G− G◦− eαG−
!
=0
(4.9)
Keeping in mind that the first matrix is N ×2N , while the second is 2N ×2N , this condition
is fulfilled if
−AT0
1 G+ = 0
and
41
1 A0 G− = 0 .
(4.10)
These requirements are satisfied by choosing
G+ =
1
AT0
!
−A0
B 1 A0 , G− =
!
1
C −AT0
1
,
(4.11)
where B, C > 0 are arbitrary symmetric positive definite matrices. This also illustrates our
earlier point that labeling the sets of nodes as + and − reflected their connection to the
number of eigenvalues of G having each sign. Thus, a CV cluster state with a bicolorable
adjacency matrix A satisfying Eq. (4.7) can be created with a TMS Hamiltonian of the form
of Eq. (4.2), with
G=
=
1
−A0
AT0
1
!
[B − A0 CAT0 ]
B
0
!
1
−AT0
!
[BA0 + A0 C]
A0
0 −C
[CAT0 + AT0 B] [AT0 BA0 − C]
!
1
.
(4.12)
For a given A, this is the most general G that satisfies Eq. (4.9), since B and C encompass
all possible rotations of the eigenvectors and scalings of the eigenvalues that preserve the
partitioning defined by Eq. (4.10). With A fixed, the freedom to choose the G that is easiest
to implement experimentally is found solely within the choices of B and C.
This is not the most general solution to the overarching problem, however. There is
no reason a priori that we should have a completely fixed A for a given CV cluster state
that we wish to create. While all QND interactions in the original formulation of CV
cluster states for QC had the same strength (Chapter 3), this is not necessary. A weighted
adjacency matrix A corresponds to variable-strength QND interactions for the edges of the
graph. This introduces squeezing and/or reversal (q → aq, p → p/a, where a is the edge
weight) to the Gaussian correction term that accumulates after each measurement. While
very low (or very high) weights would lead to difficulty resolving the quantum state after
being heavily squeezed, for weights ∼ ±1, both theoretically and practically speaking, all
of the quantum information is still preserved under single-mode measurements made on
the cluster. Allowing A to be weighted gives additional degrees of freedom to the problem,
allowing us even greater freedom in optimizing the experimental viability of the multimode
squeezing Hamiltonian used to make our cluster state.
A corollary to this result is that any multimode squeezing Hamiltonian of the form of
Eq. (4.2) that has a full-rank G generates some weighted bicolorable CV cluster state (after
appropriate single-mode phase shifts). To see this, write G in terms of its eigendecomposition
G = VνVT , where ν is a diagonal matrix of eigenvalues and V is an orthogonal matrix. Using
elementary column operations, up to a possible renumbering of the output modes, we can
always transform V into the form of the first matrix in Eq. (4.12). The target form always
exists because it is the simultaneous column-reduced echelon form for the positive and
42
negative subspaces of G and G is assumed to be full-rank. These column operations, since
they act separately on the two subspaces, can be represented by an invertible block-diagonal
matrix M acting from the right, such that
VM =
1
−A0
AT0
1
!
.
(4.13)
The transpose of this matrix, MT , acting from the left, represents the same action as row
operations on VT . With M being invertible and block-diagonal, we can choose a B, C > 0
such that
M
−1
νM
−T
=
B
0
!
0 −C
.
(4.14)
Thus, we can always write
G = VM(M−1 νM−T )MT VT
!
!
1 −A0
1
B 0
=
T
−AT0
0 −C
A0
1
A0
1
!
(4.15)
for some particular A0 . Comparing this with Eq. (4.12), we can immediately extract A0 and
use Eq. (4.7) to write A in terms of it. This completes the proof. We therefore also know
that any multimode squeezing Hamiltonian generates a weighted bicolorable CV cluster
state (generally with a different graph A) as long as the TMS adjacency matrix G is fullrank.
Intuitively, what’s happening with this correspondence is that H from Eq. (4.2) is used
to squeeze the vacuum along N joint quadratures (since G is full-rank) with overall squeezing
strength α. In general, these states are not CV cluster states because they do not satisfy
Eq. (4.1) for any choice of A in the large-α limit. What we have shown is that by partitioning
the resulting output modes into two groups (corresponding to the number of ± eigenvalues
of G) and phase-shifting one of those groups by −π/2, we can always transform the output
from the multimode squeezer into a CV cluster state, satisfying Eq. (4.1) for some choice
of A as α becomes large. Our derivation requires that A be bicolorable for this to work.
As an example, let G be the complete graph on four nodes. This generates a GHZ
state [82] whose quadrature operators satisfy q1 + q2 + q3 + q4 → 0, p1 − p2 → 0, p1 − p3 → 0,
and p1 − p4 → 0 (and any linear combinations thereof). Phase-shifting mode 1 (although
any mode will do) by −π/2 means that now −p1 + q2 + q3 + q4 → 0, q1 − p2 → 0, q1 − p3 → 0,
and q1 − p4 → 0, which satisfies Eq. (4.1) with A being the star graph on four nodes, with
node 1 in the center. This property generalizes: G being the complete graph on N nodes
creates an N -mode GHZ state, which is equivalent to an N -mode star-graph CV cluster
state after phase-shifting one of the output modes by −π/2. The shifted mode becomes the
43
OPO
optical cavity
nonlinear
crystal
pump
frequencysensitive
measurements
cluster state
Figure 4.1: Experimental schematic for single-OPO cluster state generation. A multimode
continuous-wave pump beam is incident on an optical parametric oscillator (OPO), consisting
of an optical cavity with a nonlinear crystal inside. This produces a continuous-variable cluster
state in a single beam out the other end. The output beam is in a single spatial mode,
with multimode entanglement generated between cavity resonant frequencies, so frequencysensitive measurements are needed to implement one-way quantum computing. The OPO is
operated below threshold so that the pump beam can be treated as classical, implementing
Eq. (4.2) for some two-mode squeezing matrix G [82]. Operation above threshold may also
be possible [83]. The multiple interactions required to generate a square cluster state with
graph A from Eq. (4.16) are illustrated in Fig. 4.2. If needed for other graphs, more complicated
interactions are possible through more general quasiphasematching techniques [86].
central node in the star. This mimics the case for qubits [94], although the analogy is not
exact since G and A represent different types of graphs (TMS and CV-cluster, respectively).
4.3
Experimental Proposal: Square-Graph CV Cluster State
Star graphs are not universal for QC, however. We’d like to achieve a procedure for generating a square-lattice (or other QC-universal) CV graph with a single OPO. Such a graph
is bicolorable, so a corresponding G can be constructed to create it and, in principle, can be
quasi-phase-matched in a single photonic quasicrystal [86]. A significant step in this direction is the creation of a CV cluster state with a square graph from a single four-mode OPO:
one can, indeed, show the remarkable result,
1
A0 = √
2
−1 1
1
1
!
=⇒ G =
0
A0
AT0
0
!
=A.
(4.16)
Notice that A is weighted so that one of the edges (sides of the square) has an opposite
√
interaction sign to the three others and all have magnitude 1/ 2. A (nonunique) generating G is identical and immediately implementable using current technology, in fact, using the
existing nonlinear crystal [85] designed to produce the four-party CV GHZ state (Fig. 4.1
and Fig. 4.2). Defining α as in Eq. (4.5), the variance in each of the components of p − Aq
for this state is 2e−2α units of vacuum noise. Since these vanish as α → ∞, this is a valid
square-graph CV cluster state.
4.4
Conclusion
44
Figure 4.2: Experimental implementation of a square CV cluster state using a single OPO
based on a periodically poled birefringent crystal, such as KTiOPO4 (studied in detail in
Ref. [85]). Left: the cluster graph A after a phase-shift
√ of modes 3 and 4 by −π/2; dashed
line denotes a negative weight; all magnitudes are 1/ 2. Right: the experimental proposal.
Note that the graph A on the left directly specifies the interactions G on the right only because
Eq. (4.16) allows G = A. Top arrows are OPO modes, bottom arrows are pumps, and all have
polarization directions y or z along the crystal axes. Nonlinear interactions simultaneously
phasematch (first letter is pump) yyz (open circles), yzy (filled circles), zzz (open squares),
and zyy (filled squares). The OPO cavity resonance conditions and crystal birefringence ensure
that no other OPO mode can be coupled to these four modes. Note the crucial importance of
the π-shifted pump −z.
45
In conclusion, this chapter demonstrates that any continuous-variable cluster state with a
bicolorable graph can be generated from the application of a single multimode squeezing
Hamiltonian. In addition, all multimode squeezing Hamiltonians that have a full-rank twomode squeezing adjacency matrix correspond to a weighted bicolorable continuous-variable
cluster state, generally with a different graph. While as resource-efficient as the most
efficient scheme currently known [77], these results are important for experiments because
they provide a powerfully scalable means of generating continuous-variable cluster states
using only one OPO and no beam-splitter network.
Several new concepts were introduced in this chapter. The first is the notion of a second
type of graph state called a two-mode-squeezing graph G. This graph specifies a Hamiltonian
that generates multimode entanglement between cavity modes, as in Eq. (4.2). In contrast,
a cluster-state graph A specifies the unitary that would (in the original scheme) be used to
create the cluster state from a collection of single-mode-squeezed states. The connection
between them is the main result of this chapter and is given by Eq. (4.12).
The second concept is generalizing A to include weighted graphs, which, in the original
scheme of Chapter 3, would correspond to a QND gate with varying strengths between node
pairs, e.g., exp(iAjk qj qk ). The notion of a weighted graph has unresolved implications for
error correction in this scheme. While any nonzero weights will entangle the modes in the
limit of infinite squeezing (which allows this result to hold), this is almost certainly not the
case for any realistic scheme with finite squeezing. Error correction on continuous-variable
cluster states remains an open problem, with weighted graphs adding yet another level of
complexity to it.
The method described here is most certainly “ultracompact” since all of the interactions
are performed within a single nonlinear medium inside an optical cavity. Since the proof
of correspondence between two-mode squeezing graphs G and cluster-state graphs A is
not constructive, however, it is not clear whether the method will provide any practical
advantage, since optimal implementation might still require a complicated OPO design.
The power of this method is illustrated in the next two chapters, with simple OPO and
pump beam designs proposed for a variety of useful cluster states, including, in Chapter 6,
one that is universal for continuous-variable quantum computation.
46
Chapter 5
Entangling the Optical Frequency
Comb: Simultaneously Generating
Many Small CV Cluster States
Chapter 4 established the connection between two-mode squeezing (TMS) graphs G and
continuous-variable cluster-state graphs A, given by Eq. (4.12). The freedom given by the
positive semi-definite matrices B and C makes the connection nonunique and generally quite
complicated. Furthermore, while the interactions represented by any given G are possible
in principle to (quasi)phasematch [86] within a single OPO, it still may be experimentally
difficult to do so. In this chapter, experimental and mathematical simplifications are made
in order to streamline the construction and demonstrate the power and simplicity of this
single-OPO generation method. This chapter follows Ref. [3] closely.
5.1
Multipartite entanglement in the optical frequency comb
In quantum optics, the set of resonant modes of an optical cavity, analog to the harmonics
of a vibrating string, can be seen as a set of quantum harmonic oscillators whose frequencies
are equally spaced, in the absence of dispersion, by the free spectral range (mode frequency
spacing). If a nonlinear medium is inserted in the cavity and pumped by a monochromatic
mode at frequency ωpump , pairs of modes (m, n) of respective frequencies that ωm + ωn =
ωpump (temporal phase-matching condition, or photon energy conservation) will become
coupled (Fig. 5.1). This coupling manifests itself by the creation or annihilation of a photon
pair with one photon in mode m and one photon in mode n. It is well-known that such
an OPO yields bipartite entanglement below [87, 88] and above [89, 90, 91] the OPO
threshold. The principle of the method lies in considering whether an appropriate network
of such bipartite couplings may lead to multipartite entanglement. Solutions always exist
47
Figure 5.1: Physical system and corresponding H-graph for a single pump mode. The flow
of time is from bottom to top of the figure. The horizontal direction is the optical frequency
axis for the signal OPO modes, with the pump modes denoted by half their frequency. The
H-graph is drawn below and corresponds to mere pairwise entanglement.
for GHZ states [82] and cluster states (Chapter 4), although they are not unique. In the
following, I give a detailed description of the method.
5.2
H (Hamiltonian)-graph states: physical description
The Hamiltonian under consideration is of the form of Eq. (4.2), repeated here for reference:
H = i~κ
X
Gmn (a†m a†n − am an )
(5.1)
m,n
(also see the discussion of cavity damping and entanglement generation in the paragraph
following Eq. (4.2)), where κ is an overall coupling strength, am is the photon annihilation
operator for cavity mode m, and G = (Gmn ) is a square symmetric matrix describing the
Hamiltonian coupling network: we make the hypothesis of equal coupling constants for all
coupled modes, so that the elements of G are either 0 or 1. (We will relax this later to
allow −1 as well.) The matrix G is thus the adjacency matrix of a graph representing H,
which we refer to as the H-graph as a reminder of this correspondence. An H-graph has
vertices denoting the field modes (in a vacuum state) and edges representing the non-zero
terms of H. The H-graph allows us to easily visualize the couplings associated with the
implementation of the Hamiltonian in Eq. (5.1). We sometimes refer to states created by
interactions of the form of Eq. (5.1) as H-graph states, or even simply H-graphs if this
will not lead to confusion with the actual graph itself. Fig. 5.1 displays the example of a
48
monochromatically pumped (type-I) OPO, which can only generate entangled pairs. The
H-graph adjacency matrix in this case, denoted G1 , has a constant main skew diagonal:

0 0 0 1


0 0 1 0

G1 = 
0 1 0 0 .


1 0 0 0

(5.2)
In Chapter 4, G was constrained to be symmetric, real, and full rank. In this chapter,
more emphasis is given to the experimental viewpoint and another constraint is added. Since
the optical modes are labeled by their frequencies, the temporal phase-matching relation
(ωm + ωn = ωpump ) together with the assumption of constant interaction strength yield an
adjacency matrix G with constant skew-diagonals—that is, with ones at m + n = (const.)
for each given pump mode. Any matrix with constant skew-diagonals is called a Hankel
matrix. For the sake of experimental simplicity, we therefore restrict the discussion in this
chapter to G’s of Hankel form, even though more general ones can be considered and may
also be feasible (see Chapter 4).
Throughout the rest of the chapter and later in Chapter 6, we’ll use a more compact
notation for Hankel matrices, in which we only list the entries along the top and down the
right side of the matrix, with the top-right entry set off with slashes:
G1 = [0, 0, 0/1/0, 0, 0] .
(5.3)
This shorthand provides a way to read off the experimental implementation immediately in
the case of a cavity with an evenly spaced frequency comb of modes (and no polarization
dependence): each nonzero entry in this “vector” corresponds to a pump beam. The numerical value of the entry is proportional to the amplitude of the pump (negative would indicate
a π-phase shift), and pumps corresponding to adjacent entries are apart in frequency by
twice the free spectral range of the OPO cavity.
5.3
Square-cluster OPO
Unlike qubit graph states (which include square-lattice cluster states) [94], the entanglement
properties of H-graphs have not yet been studied in detail, except for our recent existence
proof of an analytic correspondence between H-graphs and cluster states in Capter 4. It
is, however, remarkable that very simple H-graphs on the frequency comb may yield fairly
elaborate cluster states. Consider the physical system of Fig. 5.2, where two pump modes
interact in the nonlinear crystal to yield four interacting modes. Note that it is crucial that
the red connections be limited to only the one drawn in the figure. For example, if there
were a fifth mode in the cavity, modes 2 and 5 (the latter, not drawn) must not be coupled
49
Figure 5.2: Physical system and corresponding H-graph for a bimodal pump. Here, the
interactions connect four modes each. As is shown in the text, these form a square cluster
state. As in all such figures in this chapter, the flow of time is from bottom to top of the figure.
The horizontal direction is the optical frequency axis for the signal OPO modes, with the pump
modes denoted by half their frequency.
by the pump mode drawn in red. This can be achieved by engineering the nonlinear optical
interactions in quasi-phase-matched materials [86, 95], such as periodically poled KTiOPO4
(PPKTP) [85]. Referring to Fig.5.2 (and remembering that pumps are drawn at half their
true frequency), we can write down the compact form of the adjacency matrix by inspection:
G2 = [0, 0, 0/1/0, 1, 0] .
(5.4)
Under these conditions, it can then be shown that the 4-mode set actually generates a CV
cluster state with a square graph. More precisely, the defining relation for CV cluster states,
Eq. (4.1) is satisfied,
p − Aq → 0,
(5.5)
where, as before, p and q are column vectors of quadratures for each field mode, A is the
adjacency matrix for a (possibly weighted) graph, and the arrow denotes the limit of large
squeezing. Note that the source of entanglement is a single OPO. Moreover, the present
experimental proposal is simpler than that described in Section 4.3.
We’ll now examine a particular-case proof of the claim that the H-graph created by G2
corresponds to a CV cluster state. Solving the Heisenberg equations for mode evolution
50
under Eqs. (5.1) and (5.4), we obtain the following squeezed joint operators:
(−δ− q1 + δ− q2 − q3 + q4 )e−rδ+ ,
(5.6a)
−rδ−
,
(5.6b)
−rδ+
,
(5.6c)
(δ+ p1 − δ+ p2 − p3 + p4 )e−rδ− ,
(5.6d)
(−δ+ q1 − δ+ q2 + q3 + q4 )e
(δ− p1 + δ− p2 + p3 + p4 )e
√
where r = κt and δ± = ( 5 ± 1)/2. Next, we compare these to the cluster-state defining
relation, Eq. (5.5), using the adjacency matrix A given by
√ 
−2 − 5
√


 0
0
5
2 

 ,
√
A=

−2
5
0
0
 √

− 5 2
0
0

0
0
(5.7)
which corresponds to a weighted square graph with a certain choice of weightings. This
yields
√
p1 + 2q3 + 5q4
√
p2 − 5q3 − 2q4
√
p3 + 2q1 − 5q2
√
p4 + 5q1 − 2q2
→0,
(5.8a)
→0,
(5.8b)
→0,
(5.8c)
→0,
(5.8d)
which can be easily seen to be equivalent to Eqs. (5.6), in the limit r → ∞, if we rotate
modes 3 and 4 by π/2 (q 7→ p, p 7→ −q). This completes the proof.
5.4
5.4.1
Multiple square cluster states from a single OPO
Principle and first experimental implementation
As seen in Fig. 5.1, a single pump mode can generate an arbitrary number of entangled pairs
in the same OPO, at least within the phase-matching bandwidth, which can be made orders
of magnitude larger than the OPO’s free spectral range. This somewhat trivial fact can be
generalized to the square cluster of Fig.5.2: with the same number of pump modes (two),
an arbitrary number of square clusters can actually be generated, as depicted in Fig. 5.3 for
three copies. This corresponds to a Hankel matrix of the same pattern as G2 but for 4N
modes, rather than four. Once again, we can immediately read off the Hankel-shorthand
form of G3 by inspection of the diagram:
G3 = [011 /1/05 , 1, 05 ] ,
51
(5.9)
Figure 5.3: Physical system and corresponding H-graph for a bimodal pump. Here, the
interactions define three (or more) sets of four modes each. As is shown in the text, each of
these form a square cluster state. As in all such figures in this chapter, the flow of time is from
bottom to top of the figure. The horizontal direction is the optical frequency axis for the signal
OPO modes, with the pump modes denoted by half their frequency.
where 0n is shorthand for a string of n zeros. From an experimental perspective, the few
pump modes and single type of nonlinear interaction greatly simplify the setup. One still
has to prevent the red pump from introducing spurious mode couplings (i.e., couplings
of the indicated modes to other modes outside the set), for example by designing sharp
quasiphasematching cutoffs in the nonlinear crystal or by filtering mode amplitudes or
engineering dispersion in the OPO cavity. Such a constraint isn’t needed for the green
pump because any spurious couplings generated would be strictly outside of the desired
mode set.
5.4.2
Second experimental implementation
Note that this experimental generation of multiple square clusters is far from unique and
can be even more efficiently and conveniently implemented by use of the polarization degree
of freedom, as depicted in Fig. 5.4. This yields several advantages: first, one uses but a
single pump mode; second, one has the advantage of connecting entangled modes in closed
sets, without any additional engineering of the nonlinear interaction bandwidths. The
interactions yield exactly the same equations as Eqs. (5.8).
The experimental implementation is relatively straightforward: one can use one ZZZ
crystal placed in sequence with a Y ZY crystal (first letter denotes pump polarization and
other two the signal polarizations), with one crystal rotated at 90 degrees from the other.
In this case, a vertically polarized pump generates the necessary V HV , V V H, and V V V
interactions. Moreover, if the crystals have the same length, the fact they are rotated by 90
52
Figure 5.4: Physical system and corresponding H-graph for a single pump mode. The letters
H and V denote the optical polarizations of each mode, which are frequency degenerate. Here,
the interactions define three (or more) sets of four modes, each of which form a square cluster
state. As in all such figures in this chapter, the flow of time is from bottom to top of the figure.
The horizontal direction is the optical frequency axis for the signal OPO modes, with the pump
modes denoted by half their frequency.
53
Figure 5.5: Physical system and corresponding H-graph for two frequency degenerate pump
modes. Here, the interactions define three (or more) sets of four modes, each of which form
a square cluster state (Section 4.3). Note that the opposite interaction (dashed line) is not
necessary but gives a more balanced cluster state. As in all such figures in this chapter, the
flow of time is from bottom to top of the figure. The horizontal direction is the optical frequency
axis for the signal OPO modes, with the pump modes denoted by half their frequency.
degrees from each other ensures that the frequency combs corresponding to each polarization
will have the same free spectral range, which in turns ensures common resonance conditions
at a given cavity length.
5.4.3
Third experimental implementation
A variant of the previous implementation allows for generation of multiple copies of the exact
proposal in Chapter 4, which yields a more balanced square cluster state. This is achieved by
adding an additional interaction (Fig. 5.5, dashed lines) of opposite effect to the three former
ones (i.e., upconverting if the three are downconverting, or vice versa). This can be ensured
by seeding the OPO below threshold with coherent-state signals, appropriately phase-shifted
with respect to the pump fields. The experimental implementation consists, for example, in
using 2 PPKTP crystals: one that simultaneously quasiphasematches the Y ZY and ZZZ
interactions [85] and one only the ZZZ interaction, rotated at 90 degrees from the other.
In this case, vertically and horizontally polarized pumps generate interactions V HV and
V V H, V V V , and the opposite HHH.
54
The resulting lower weighting can be seen, as before, by solving the quantum evolution
for the multimode squeezing, to be compared to Eqs. (5.6),
(q1 + q2 −
(q1 − q2 +
(p1 + p2 +
(p1 − p2 −
√
√
√
√
2 q4 )e−r
2 q3 )e−r
2 p4 )e−r
2 p3 )e−r
√
√
√
√
2
,
(5.10a)
2
,
(5.10b)
2
,
(5.10c)
2
,
(5.10d)
and deducing, as before, the following cluster state relations, to be compared with Eqs. (5.8):
√
√
√
√
2p1 + q3 − q4 → 0 ,
(5.11a)
2p2 − q3 + q4 → 0 ,
(5.11b)
2p3 + q1 − q2 → 0 ,
(5.11c)
2p4 − q1 + q2 → 0 .
(5.11d)
These implementations are currently underway at the University of Virginia.
These methods therefore yield an arbitrary number of disconnected 4-mode square cluster states from just a single OPO, limited only by the number of modes within the phasematching bandwidth. The whole frequency comb can thus be partitioned into entangled
subsets by using only one or two pump modes. The eventual goal, of course, is an arbitrarily
large square-lattice grid (or other universal graph),1 and many disconnected squares isn’t
the same at all. Still, the fact that only one or two pumps are needed to generate so many
copies of this (rather modest) cluster state leaves us hopeful that this is a worthwhile line
of research to pursue. In the next section, we give the theoretical basis for this result and
show that this process can also be used for eight-mode cubic cluster states. These can each
be converted, if desired, into 2 × 3 square-lattice cluster states at the expense of homodyne
detection on two modes per cube.
5.5
Simplified relationship between H-graphs and clusterstate graphs
The relation between a bipartite CV cluster state, represented by the adjacency matrix
A=
1
0
A0
AT0
0
Chapter 6 achieves this.
55
!
,
(5.12)
and the aforementioned H-graph matrix G is given by Eq. (4.12), repeated here for reference:
G=
[B − A0 CAT0 ]
[BA0 + A0 C]
!
[CAT0 + AT0 B] [AT0 BA0 − C]
.
(5.13)
where B and C are arbitrary symmetric positive-definite matrices. This relation allows for
abundant (in fact, excessive) freedom in the choice of G for a given A. For concreteness,
let’s take B = C = 12 , where 1 is the identity matrix. Now assume that A is orthogonal, i.e.,
A0 AT0 = AT0 A0 = 1. Then Eq. (5.13) yields
G=
0
A0
AT0
0
!
=A.
(5.14)
Restricting to orthogonal A’s therefore can be used to eliminate the distinction between CV
graphs and H-graphs, simplifying the problem greatly (at the expense of some generality).
Keeping in mind the possible limitations of such a restriction, we’ll focus now only on
orthogonal A’s. Using ∼
= to indicate equality up to a renumbering of modes, we can then
inquire whether we can always find a G ∼
= A, where G is in Hankel form, for obvious
experimental convenience.
5.6
Simultaneously generating multiple copies of a CV cluster state
For reasons that will become clear shortly, let’s begin with a bipartite graph having a
orthogonal, Hankel A0 . Since A0 is Hankel, it is automatically symmetric. Thus, Eq. (5.12)
can be expressed as A = F2 ⊗ A0 , where
Fn = [0n−1 /1/0n−1 ]
(5.15)
is the n × n skew-identity. Since A0 is orthogonal, by the argument above, we can choose
for our H-graph,
G = A0 ⊗ F2 ∼
= F2 ⊗ A0 = A ,
(5.16)
since exchanging the order of a tensor product is equivalent to a renumbering of the modes.
It’s straightforward to show that since A0 is Hankel, then A0 ⊗ Fn (for any n) is also Hankel.
Considering that F2N ⊗ A0 is zero except for 2N blocks of A0 on the main skew-diagonal, it
can be seen that it is the adjacency matrix for N distinct copies of the graph corresponding
∼ A0 ⊗ F2N , the
to A. Also, F2N ⊗ A0 is orthogonal if A0 is orthogonal. Since F2N ⊗ A0 =
56
latter being Hankel if A0 is Hankel, then for any orthogonal, Hankel A0 , we know that
GN = A0 ⊗ F2N
(5.17)
is Hankel and will create N copies of the CV graph A as defined in Eq. (5.12).
5.7
Simultaneous generation of 2 × 2 and 2 × 3 cluster states
To apply these results, let’s first consider the case of multiple square clusters in a single
OPO. A orthogonal A0 for a single square cluster was given in Eq. (4.16):
1
A0 = √
2
−1 1
1
!
1
1
= √ [−1/1/1] ,
2
(5.18)
which is also a Hankel matrix. Using the results above, we have that
1
GN = A0 ⊗ F2N = √ [0̄, −1, 0̄/1/0̄, 1, 0̄] ,
2
(5.19)
where 0̄ = 02N −1 , is a Hankel matrix for the H-graph that will generate N disconnected
squares from a single OPO using just three pumps.
Note that the negative matrix elements in GN are required for unitarity but are absent
in Eq. (5.9) in the experimental section, which yields a CV cluster state with the same
graph but different weights. As we previously indicated, requiring that A be orthogonal
is not necessary; we do it here only for the convenience of the calculations. It proves
useful, though, because we can extend this method to generate an arbitrary number of
disconnected 8-mode cubic cluster states, each of which can be reduced by measurement to
a 2 × 3 square-lattice cluster state, if desired. We show this presently.
The orthogonal form of A0 for a cubic cluster is
1
A0 = √ [−1, −1, 1/0/1, 1, −1] .
3
(5.20)
The corresponding graph is depicted in Fig. 5.6. Note that a cubic cluster state reduces to
a 2 × 3 one under the measurement of two neighboring vertices in the position (amplitude
quadrature) basis. Using the results above, a generating H-graph matrix in Hankel form
for N disconnected cubes is
GN = A0 ⊗ F2N
1
= √ [0̄, −1, 0̄, −1, 0̄, 1, 0̄/0/0̄, 1, 0̄, 1, 0̄, −1, 0̄] ,
3
57
(5.21)
Figure 5.6: Cubic cluster-state graph. Dashed lines indicate negative-weight edges. Measurement of any two neighboring vertices (open circles) in the position (amplitude quadrature)
basis q disentangles those vertices from the rest of the graph, thereby yielding a 2 × 3 cluster.
where, again, 0̄ = 02N −1 . We stress that this generation of an arbitrary number of disconnected cubes can be achieved using only six pumps and a single OPO. The maximum
number of cubes is limited only by the number of modes within the phase-matching bandwidth.
5.8
Conclusion
In this chapter, we have expanded upon the results presented in Chapter 4 on the generation of CV cluster states with a minimum amount of physical resources, in particular by
adding the constraint that the H-graph adjacency matrix be of Hankel form and orthogonal.
Although justifiably limiting the generality of the treatment, this constraint makes it easier
to connect to concrete and simple experimental cases, for which we show that large sets of
independent 2 × 2 and 2 × 3 square-grid cluster states can be obtained in a single OPO.
Multiple single-cube cluster states can also be produced in this fashion.
A remarkable feature of this sort of Hankel-orthogonal duplication procedure is that
the number of pumps is independent of the number of squares or cubes being generated,
as is the case for multiple entangled pairs; we need only three pumps at most (and as
few as one, as shown in the experimental sections) to generate multiple squares and six
pumps to generate multiple cubes. This is significantly better than our earlier estimate
of the O(N 2 ) pump fields needed to create a cluster with N modes. This estimate was
based on the conservative assumption of one pump per graph edge and N (N − 1)/2 ∼ N 2
edges in a complete graph. We observe that square-lattice cluster states are significantly
sparser, with O(N ) edges. In addition, single pumps can couple many mode pairs when
Hankel matrices are used. Work is currently underway at the University of Virginia to
experimentally generate these CV cluster states and to explore what other types of states
can be created under these conditions.
58
This chapter takes the main result from the last chapter to a new level, showing that
single-OPO generation of continuous-variable cluster states has significant prospects for
scalability. This is demonstrated quite conclusively by the ability to generate an arbitrary
number of small cluster states with no change to the pump beam complexity. Being disconnected, the multiple copies of small cluster states cannot be considered a universal family
of cluster states because the size of each is not scaled up—only the number of copies. Still,
this chapter describes three essential advances in the method.
The first is restriction to graphs that have a Hankel, orthogonal adjacency-matrix representation. The Hankel constraint guarantees that each individual pump frequency generate lots of pairwise squeezing interactions at once (from the phasematching condition
ωp = ωm + ωn ). The second is the orthogonality constraint, which allows B = C = 1/2
in the connection, Eq. (5.13), between G (previously called a two-mode squeezing graph;
here called an H-graph) and A, allowing G = A. This restriction makes the complicated
mathematical connection between the two graphs a non-issue. The final advance introduced
in this chapter is that polarization can further simplify the pump frequency content from
a polarization-free description, allowing each individual pump frequency to generate several interactions at once. These three ingredients: Hankel form of G, orthogonality of G
(and thus, also of A), and simplification through the use of polarization are used in the
next chapter to illustrate an extremely scalable method for generating continuous-variable
cluster states of very large size.
59
Chapter 6
One-Way Quantum Computing in
the Optical Frequency Comb
This chapter, which expands on Ref. [4], is the culmination of my work on scalable implementation of continuous-variable cluster states. As such, a more thorough introduction
from that work will be retained here for the sake of a self-contained presentation and in
spite of the overlap with material presented previously.
6.1
Introduction
Quantum computing (QC) is a fascinating endeavor, ripe with promises of exponential
speedup of particular mathematical problems such as quantum system simulation [26] and
integer factoring [15]. Because implementing QC requires exquisite control of each single
quantum memory unit (e.g., qubit) in a large-size register, practical QC is therefore faced
with the daunting challenges of overcoming decoherence and achieving scalability [37]. Recently, the invention of one-way quantum computing introduced a new paradigm for quantum information processing [38] based on teleportation alone [96]. In the traditional “circuit” QC model [22], physical quantum systems carry quantum information and undergo
controlled unitary evolution; in the one-way QC model, quantum information exists virtually in a cluster state [39] and is manipulated through a sequence of local measurements.
The choice of measurement basis and the measurement results fully determine the quantum
algorithm.
The appeal of one-way QC is that it consolidates most of the challenging work into creating the universal resource—the cluster state—and that it only requires local measurements.
In addition, some one-way QC schemes admit very high fault-tolerance thresholds [44], and
experimental realizations with four qubits have already been achieved [46, 47]. Efficient
methods of creating large-scale cluster states are still needed, however, for practical implementation to be realistic. Here, we describe a radically new approach to scalability: a
60
“top-down” method to produce large continuous-variable (CV) cluster states using a compact experimental setup. The interest of CVs is their natural implementation by squeezing
(quantum noise reduction) in quantum optical systems. Photons are also less prone to decoherence than, say, atoms, due to their lower propensity to interact with the environment.
Several studies have established the use of photonic CVs for teleportation [97, 98], QC [52],
quantum error correction [56], cluster states [66, 78, 79] (plus Chapters 3, 4, and 5), and
one-way QC (Chapter 3).
6.2
CV cluster states from a single OPO
Our implementation of a scalable one-way quantum processor uses one multimode optical
parametric oscillator (OPO). Initial proposals for constructing CV cluster states [66] (also,
Chapter 3) involved inline squeezers (seeded OPOs) [76], which are difficult to implement.
A more viable method relies on the Bloch-Messiah decomposition [72] and uses N vacuum
squeezers followed by an O(N 2 )-port interferometer [77]. Our method improves further,
requiring only a single OPO and no interferometer (Chapter 4), and provides huge scaling
potential.
The OPO combines two essential elements. The first one is an optical cavity (e.g.,
two facing mirrors) whose spectrum of resonant frequencies forms an optical frequency
comb (OFC), so called because of the equal spacing between modes. Considered as a
quantum system, the OFC is a large collection of independent quantum harmonic oscillator
modes, or “qumodes” (a term used in analogy with “qubit”). Quantum information is
encoded in the quadrature field variables of the OFC, which are analogs of position and
momentum for a mechanical oscillator. The OFC can comprise millions of modes and
has outstanding classical coherence properties that have found groundbreaking applications
in the revolutionary and now ubiquitous use of mode- and carrier-envelope-phase-locked
femtosecond lasers in time/frequency metrology [80, 81].
The second crucial element of the OPO is the (largely) nondissipative nonlinear medium
placed in the cavity. In a basic OPO, this medium is pumped by a monochromatic field
and promotes downconversion, i.e., the simultaneous absorption of a pump photon at
frequency ωp and emission of a photon pair at OFC frequencies ωm and ωn such that
ωp = ωm + ωn , as well as upconversion, the reverse process. Such interactions yield bipartite CV entanglement of the OFC qumodes at frequencies ωm and ωn [99]. In the OPO we
propose to use, the nonlinear medium is specifically engineered to quasiphasematch [95, 100]
several such interactions simultaneously [85]. This, along with a polychromatic pump, allows one to “write” an entangled network onto the OFC using pairwise couplings. We show
in this paper that this network can be made to precisely constitute a large CV cluster
state, universal for one-way QC, and whose scaling requires no increase in pump complex-
61
ity (number of frequencies) and only linear increases in pump intensity and nonlinear gain
bandwidth.
As in previous chapters, a CV cluster state is defined as any member of a family of
squeezed states indexed by an overall squeezing parameter r > 0 for which the variance
of each component of (p − Aq) tends to 0 as r → ∞ [77]. Here, q = (q1 , . . . , qN )T and
p = (p1 , . . . , pN )T are vectors of amplitude and phase quadrature operators, respectively,
and A is the (weighted, undirected) adjacency matrix of the cluster state’s N -node graph
(Chapter 4). The infinite-squeezing limit is not achievable by any finite-energy state, an
important point we address later.
We consider an OPO that implements the following Hamiltonian, in the interaction
picture and assuming a classical undepleted pump [82]:
H(A) = i~κ
X
Amn (a†m a†n − am an )
(6.1)
m,n
(also see the discussion of cavity damping and entanglement generation in the paragraph
following Eq. (4.2)), where κ > 0 is an overall nonlinear coupling strength (squeezing
parameter per unit time). The nodes of the graph described by A correspond to OPO
qumodes and any (m,n) edge is weighted by Amn , whose magnitude is the qumode coupling
strength in units of κ and whose sign indicates downconversion if positive and upconversion
if negative. In Chapter 4, we saw that H(A) generates a CV cluster state whose graph is
generally not given by A. If A is an orthogonal matrix, though, the graph is given by A—a
fact used in Sec. 5.7 to construct an H(A) that generates large sets of very small (2 × 2 or
2 × 3) CV cluster states. Being disconnected, these cluster states are not universal for QC,
but requiring orthogonality is a useful simplification. Here we construct an orthogonal A
for which H(A) generates a QC-universal CV cluster state.
Labeling rows and columns of A by sequential OFC modes, H(A) is easy to implement
experimentally when A is a Hankel matrix (i.e., has constant skew diagonals). Any pump
frequency ωp satisfying ωp = ωm +ωn , together with the assumption of a constant interaction
strength, generates a constant skew diagonal in A and sets all its entries to the same value,
fixed by the pump power. Additional pump frequencies generate additional skew diagonals,
resulting in A having Hankel form. This connection, along with a useful shorthand for
Hankel matrices, is illustrated1 in Figure 6.1.
6.3
Single Mode-Universal CV Cluster State
We first illustrate the main ideas on a simpler graph that is universal for single-mode
operations (but not for universal QC, which requires multimode operations). We desire such
1
At the time of this writing, a supplemental animation made by my coauthor, Olivier Pfister, can be
found at http://faculty.virginia.edu/quantum/torus.mov .
62
Figure 6.1: Hankel shorthand and pump specification. A Hankel matrix is uniquely specified
by the entries along the top and down the right side. We collect these entries into a shorthand
vector, representing the entire matrix itself, with the top-right entry set off with slashes. When
A is Hankel, its shorthand vector immediately specifies the pump spectrum required to implement H(A). Each nonzero entry in the shorthand vector denotes the amplitude of a frequency
in the pump, each of which generates CV entanglement between pairs of qumodes in the OFC
symmetric about half that frequency. This accounts for all couplings prescribed by A.
a CV cluster state, which also has a bicolorable graph (Chapter 4), and whose adjacency
matrix is orthogonal and Hankel for experimental simplicity (Chapter 5). Orthogonality of
the adjacency matrix (AAT = 1) for an undirected graph (A = AT ) yields A2 = 1, or
(A2 )jk =
X
Ajl Alk = δjk .
(6.2)
l
Eq. (6.2) has a geometric interpretation: (An )jk represents the sum of the weights of all
n-length paths from node j to node k, where the weight of such an “n-path” equals the
product of all edge weights along the path. Eq. (6.2) enforces two conditions:
(1) all 2-paths that begin and end on the same node have weights that sum to 1, and
(2) all 2-paths that link different nodes have weights that cancel out.
The simplest graph to try would be a line graph [38] but such a graph necessarily fails
condition (1) because of irregularity at the boundaries. The next natural graph to consider
is therefore a ring graph, which is a regular graph, but orthogonality is still prohibited
because each node is connected to its next-nearest neighbor by only one 2-path,2 for which
the sum in Eq. (6.2) collapses to a single term (which must be nonzero), thus violating
condition (2).
2
The exception is a four-node ring. This is unimportant since we want a scalable procedure that works
for a large number of nodes.
63
a
=
b
c
d
Figure 6.2: Interpretation of a matrix-weighted edge. A matrix-weighted edge connecting
two macronodes specifies the graph connecting the underlying physical nodes. In this example,
the macronodes are connected by a 2 × 2 matrix-valued weight (left). Each macronode itself
therefore contains two physical nodes, and the real-valued weights connecting them are specified
by entries in the matrix (right).
While real-valued weights cannot satisfy Eq. (6.2) for a ring graph, matrix-valued weights
can, as shown below. In such a case, Eq. (6.2) becomes
X
Ajl Alk = δjk 1 ,
(6.3)
l
where the “entries” Ajl are themselves m × m matrices. This means A is now an adjacency
matrix on an (mN )-node graph. On the other hand, treating the m × m blocks as single
entries, A is the matrix-weighted adjacency matrix for what we call a supergraph, which has
macronodes consisting of m individual nodes each. Figure 6.2 illustrates the meaning of
matrix-weighted edges between two macronodes.
Promoting the ring graph to a supergraph with 2 nodes per macronode (i.e., m = 2), we
can choose our weighting to consist of two orthogonal projectors over R2 , labeled π ± , with
Aj,j+1 = π + for j even and Aj,j+1 = π − for j odd, where the total number of macronodes N
is even, and index addition is modulo N . By reasons of symmetry and connectivity of the
underlying graph, we choose π ± to be
1
π+ =
2
!
+ +
+ +
and
1
π− =
2
!
+ −
− +
,
(6.4)
with ± standing for ±1. With these weights, the supergraph is still a ring, but the actual
graph on physical nodes has a more complicated “crown” structure, as shown in Figure 6.3.
Either by direct verification of Eq. (6.3) or by noting that the geometric conditions (1)
and (2) are satisfied, this weighting of the ring supergraph results in an orthogonal A.
Given a CV cluster state with this graph, one can perform local measurements on one
of the two rings of physical nodes and measure it down to a CV cluster state (Chapter 3)
having a simple ring graph with equal weights, which is universal for single-mode operations
64
Figure 6.3: Matrix-valued weights and supergraphs. The matrix-valued weights π ± , defined
in Eq. (6.4), connect the macronodes of a “ring” supergraph (left). The entries in π ± specify the
real-valued weights in the actual “crown” graph (right) that connects the underlying physical
nodes. Measuring q for each of the physical nodes in the top layer of the crown leaves the
bottom layer in a uniformly weighted ring-graph CV cluster state.
using one-way QC methods [38, 75]. The only thing remaining is to show that the adjacency
matrix for the crown can be put into Hankel form by an appropriate numbering of the nodes.
It is easiest to begin at the supergraph level. The ring is a circulant supergraph, which
means it can be represented by a circulant adjacency matrix simply by numbering the nodes
sequentially around the ring. What we want, however, is the skew-circulant form, a special
case of Hankel where the skew-diagonal bands “wrap around” at the edges of the matrix.
We can renumber the nodes to achieve this form as follows. Starting with an arbitrary
node, label it 1. Move two nodes clockwise, and label that node 2. Continue labeling every
other node sequentially in a clockwise fashion until you reach node N/2. Then, pick one
of the remaining nodes to label as N/2 + 1, move two nodes counterclockwise, and label
that node N/2 + 2, continuing in this fashion until all nodes are labeled. The result is a
skew-circulant adjacency matrix at the supergraph level, with matrix-valued weights given
by Eq. (6.4). Such a matrix is block-skew-circulant (and thus block-Hankel) at the physical
node level:
A∼
= [0, . . . , 0, π + , 0/π − / 0, . . . , 0, π + , 0] ,
| {z }
| {z }
N −3
(6.5)
N −3
where the zeroes are 2 × 2 blocks of all zeros, and ∼
= indicates equality up to renumbering
of nodes.
We can use the following method to renumber the individual nodes so that A is fully
Hankel. Since A is an N × N matrix of 2 × 2 blocks, any index j can be thought of as an
ordered pair [m, c] := 2m + c = j, with m ranging from 0 to N − 1 and c being either 0 or 1.
The permutation that reverses the order of the subindices, [m, c] → [c, m] := N c + m = j 0 ,
65
can be used to convert A into fully Hankel form:
1
A∼
= [0N −3 , 1, 0, 1, 0N −3 , 1, 0/−1/0N −3 , 1, 0, 1, 0N −3 , 1, 0] ,
2
(6.6)
where 0n represents a string of n zeros. Being fully Hankel, Eq. (6.6) can be used to
directly read off the seven pump frequencies needed to implement this CV cluster state in
a single OPO using an optical frequency comb of qumodes by the correspondence shown in
Figure 6.1. Notice that since the only effect of increasing N is to add more zeros to this
Hankel vector—shifting, but not increasing in number, the required pump frequencies—the
complexity of the pump is constant with respect to N , making this construction extremely
scalable.
The CV cluster state described above therefore fulfills the four criteria desired in this
section: it is bipartite, it has an orthogonal adjacency-matrix representation, that representation is also Hankel, and it can be used to implement any single-mode operation. The first
property satisfies the necessary condition for generation in a single OPO (Chapter 4). The
second satisfies our desire for mathematical simplicity, allowing us to use the cluster-state
graph A without modification as the H-graph from Eq. (6.1). The third guarantees simple experimental implementation (Chapter 5), with constant pump beam complexity. The
fourth says that single-mode measurements can be used to effect any single-mode unitary
operation. While not universal for multimode QC over continuous variables, the construction described here has introduced all the essential concepts for a construction that achieves
this goal, described next.
6.4
QC-Universal CV Cluster State
The construction in the previous section gave a spectacularly efficient method of generating a
CV cluster graph with one-dimensional topology (such as a line or ring), hence only suitable
for single-mode operations. In universal one-way QC, however, operations involving more
than one qubit/qumode require graph connections between such one-dimensional “quantum
wires.” The natural candidate for a fully universal CV cluster-state graph is the square
lattice—the original graph proposed for qubit cluster states [38]—but, similar to the case
made against the line graph, a square lattice does not admit an orthogonal adjacency matrix
because it is not regular at the boundaries. A logical response is then to impose toroidal
boundary conditions, linking nodes on one side to those on the opposite side. However, the
resulting toroidal square lattice suffers from the same problem as the simple ring: there exist
pairs of nodes connected by exactly one 2-path, prohibiting an orthogonal representation.
We therefore take the same measures and promote the toroidal lattice to a supergraph.
Recall that orthogonality requires that A2 = 1, equivalent to the aforementioned two geometrical conditions (1) and (2), which are local on the (super-)graph and thus are not
affected by the overall topology. In this case, the supergraph locally looks like a square
66
Figure 6.4: Four-color solution to the geometric orthogonality conditions. Labeling the edges
of a large lattice with four colors as shown, where each color represents one of four orthogonal
projectors over R4 , satisfies the geometric orthogonality requirements (see text). From each
node protrudes exactly one edge of each color, satisfying condition (1), and any 2-path connecting distinct nodes (several shown) traverses edges with two different colors, which has exactly
zero weight, satisfying condition (2).
lattice, which has degree four, and Figure 6.4 illustrates that four mutually orthogonal
projectors are sufficient to satisfy the geometric conditions. Once again motivated by symmetry and connectivity of the underlying graph, we choose the weights to be 4×4 projectors
constructed from the 2 × 2 projectors in Eq. (6.4):
0
+
+
Π =π ⊗π =
Π2 = π − ⊗ π + =
+ + + +
+
+
+
+
+
1
−
4
−
1
4
+
+
+
+
+
−
−
+
+
+
−
−
+
+
+
+
+
−
−
+
+
1
+
−
,
Π =π ⊗π =
,
Π3 = π − ⊗ π − =
+ − + −
−
+
−
+
−
1
−
4
+
1
4
+
−
+
−
+
+
−
−
+
−
−
+
+
−
+
−
+
+
−
−
+
,
.
(6.7)
We now possess an orthogonal supergraph, with 4-node macronodes, that looks locally like a
square lattice but has toroidal topology. The final task is to convert the resulting adjacency
matrix into Hankel form.
Some freedom remains in the way that the lattice is rolled up into a torus that will
turn out to make a Hankel form of the supergraph particularly easy to implement. The
freedom lies in the fact that we can twist the lattice as we roll it up: instead of connecting
the last node of each line of the lattice to the first node in that same line, we can instead
connect it to the first node of a different line—one that is n lines away from the original
one—generating a twist of n lines in the torus. Finding a Hankel form for the ring relied
on the fact that the graph is circulant, along with a suitable node renumbering procedure,
67
to generate a skew-circulant adjacency matrix. While we could start with a toroidal lattice
and try to find a twist that admits a Hankel form for it, we will instead explicitly construct
a circulant graph and show that it is in fact a twisted toroidal lattice. We will construct
the graph from two different circulant “threadings” of M 2 nodes, corresponding to the two
different dimensions of the lattice.
The simplest circulant threading is just an M 2 -node ring graph (we require M to be
even). Alternating between two of the projectors from Eq. (6.7), say Π2 and Π3 , this is
exactly the ring supergraph from the previous section (except now with 4-node macronodes).
We then apply our second “threading” by additionally connecting each node in the ring to
the two nodes that are M + 1 steps away, alternating with the remaining two projectors,
in this case Π0 and Π1 (see Figure 6.5). Since M + 1 and M 2 are relatively prime, this
procedure creates another complete cycle through the ring, this time traversing every node
by taking steps of size M + 1. The result is a circulant graph where each node now has
four neighbors. While tracing any path through the nodes, each step will move in one of
the two threading “directions,” moving by ±1 or ±(M + 1). These correspond to moving
horizontally or vertically, respectively, through the lattice. The circulant nature of the
graph guarantees toroidal boundary conditions. Thus, we are left with a twisted toroidal
square lattice supergraph, shown in Figure 6.6.
Since this graph is circulant, we can use the ring renumbering trick from the previous
section (with N → M 2 ) to generate a skew-circulant adjacency matrix for this supergraph,
which is block-skew-circulant (and thus block-Hankel) with 4 × 4 blocks at the level of the
physical nodes:
A∼
= [0u , Π1 , 0v , Π0 , 0u , Π3 , 0/Π2 /0u , Π1 , 0v , Π0 , 0u , Π3 , 0] ,
(6.8)
where u = (M −1) and v = (M 2 −2M −3). Once again, the resulting matrix is block-Hankel
instead of fully Hankel, this time with 4 × 4 blocks. For the ring, we used a renumbering
of the nodes to convert the 2 × 2-block-Hankel matrix from Eq. (6.5) into the fully Hankel
matrix in Eq. (6.6). An equivalent renumbering can be done in this case to convert this 4×4block matrix to one with 2 × 2 blocks, which almost preserves the Hankel structure (we’ll
show how to fix it below): simply treat the 4 × 4 blocks of A as 2 × 2-block matrices of 2 × 2
blocks, and apply the renumbering from the previous section (this time, with N → 2M 2 ).
The resulting matrix is not completely block-Hankel, but it is close. We can fix this by
defining A0 as in Eq. (6.8) but with the sign flipped in the next-to-last entry (Π3 → −Π3 ).
While A is block-skew-circulant, A0 is only block-Hankel (but still orthogonal):
A0 = [0u , Π1 , 0v , Π0 , 0u , Π3 , 0/Π2 /0u , Π1 , 0v , Π0 , 0u , −Π3 , 0] ,
68
(6.9)
Figure 6.5: Circulant embedding of a twisted toroidal lattice. The 16 white spheres each represent a four-node macronode, while the colored edges each represent a matrix-valued weight Πj
from Eq. (6.7), with Π0 red, Π1 yellow, Π2 blue, and Π3 green. The toroidal “axes” are identified with moving along the “red-yellow direction” and the “blue-green direction.” The circulant
construction includes a one-unit twist in each toroidal dimension and generalizes easily to more
nodes (see text).
69
70
Figure 6.6: Toroidal lattice supergraph and underlying graph structure. Each of the 222 macronodes in the supergraph (left) consists of four
physical nodes, and each color corresponds to one of the four matrix-valued weights Πj from Eq. (6.7). Entries in Πj specify the real-valued
weights connecting the underlying physical nodes (right). Measuring q on each physical node in three of the four “layers” leaves the remaining
layer in a uniformly weighted QC-universal toroidal lattice cluster state.
The important thing, though, is that if we apply the renumbering procedure to A0 , the
result is a 2 × 2-block-Hankel adjacency matrix:
A0 ∼
= [0s , π − , 0t , π + , 0s , π + , 0, π − , 0s , π − , 0t , π + , 0s , π − , 0/π + /
0s , π − , 0t , π + , 0s , π + , 0, π − , 0s , π − , 0t , π + , 0s , π − , 0] , (6.10)
where s = (2M − 1) and t = (M 2 − 4M − 3).
The M -indexed family of these cluster states is universal for CV one-way QC. To see
this, first cut open the toroidal lattice as shown in Figure 6.7, to form an ordinary lattice.
Then, measure q on three physical nodes per macronode to reduce the supergraph to a
uniformly-weighted graph with the same structure (see Figure 6.6), which is known to be
a universal graph (Chapter 3). These cluster states are QC-universal, bicolorable, and orthogonal —but still only block-Hankel. This is nevertheless sufficient for simple experimental
implementation, using a method we now describe.
6.5
Experimental implementation
In Eq. (6.10), each π ± block corresponds to a single pump frequency. Such couplings can be
implemented using the two orthogonal polarizations of an optical field at a given frequency,
as was experimentally demonstrated by simultaneously quasiphasematching polarizationsensitive interactions ZZZ, ZYY, and YZY/YYZ (first letter is pump polarization) in a
periodically poled KTiOPO4 (KTP) crystal [85]. The difference between π − and π + is
thus a 180◦ phase-shift in the Y-polarized pump mode. A narrowband pump polarized
at ±45◦ in the (ZY) plane implements a π ± skew-diagonal band in A. Equation (6.10)
therefore translates into a single OPO pumped by exactly 15 frequencies. Interactions with
OFC modes outside the desired subset must also be strictly suppressed, which can be done
by cavity mirror design and/or by quasiphasematching bandwidth design [95]. While the
pump spectrum is relatively complex—requiring 15 frequencies—that number is constant
with respect to the lattice size, making this construction extremely scalable.
Note that, in principle, additional physical parameters like wave vector direction or
transverse mode structure could be used to directly implement block-Hankel A’s with larger
blocks (e.g., 4 × 4). This would reduce pump complexity but require a more sophisticated
OPO.
Finally, we provide realistic estimates for the scaling potential of the CV cluster state
with N macronodes (N = M 2 ) and constant overall coupling strength κ. The pump spectrum complexity and the OFC are independent of N . Only two quantities scale linearly
with N : the overall pump power and the bandwidth of the nonlinear coupling. The number of pump photons must increase with the number of entangled qumode pairs (number
71
Figure 6.7: Unrolling the torus. The toroidal lattice in Figure 6.6 can be considered as a
particular identification of macronodes on an infinite planar lattice that has the local structure
of Figure 6.4. The identification is specified by the (nearly) rectangular region delimited by
black borders, tessellating the entire plane. Macronodes at the same location with respect
to their containing region are considered to be the same macronode; this enforces the toroidal
boundary conditions. Each region has a one-unit “foot” sticking out to account for the one-unit
twist in each dimension (without the twist, the region would simply be square). The region
is (M + 1) × (M − 1) + 1 macronodes in size (with the +1 accounting for the “foot”); the
23 × 21 + 1 region shown corresponds to the 222 -macronode lattice in Figure 6.6. Measuring q
on each node of the macronodes highlighted red “unrolls” the torus into an ordinary lattice of
size M × (M − 2), highlighted blue. The M -indexed family of such lattices is universal for QC;
thus, the original toroidal lattice is, as well.
72
of graph edges), which grows linearly with N in a square lattice. Two-mode CV entanglement can be obtained at a few milliwatts pump power, with the upper limit being the
optical damage limit, which in KTP is at least several watts (focused, continuous-wave),
thus yielding three orders of magnitude of scaling range. The bandwidth of the nonlinear
coupling (100 GHz to 1 THz) must encompass the whole desired set of qumodes, separated by the cavity’s free spectral range (100 MHz for a 1.5-meter-long cavity), which yields
three or four orders of magnitude. These figures reflect ordinary—rather than state-of-theart—performance and do not account for other interesting avenues such as implementing
nonlinear couplings in slow light media. These estimates indicate our approach has a quite
realistic potential for significant scaling.
6.6
Finite squeezing and CV fault tolerance
The finite-squeezing approximation is a special case of more general considerations of error
correction and fault tolerance for one-way QC using CV cluster states. Certainly, more
squeezing is preferable to less, but the amount required for any particular QC task remains
an open question. Consequently, it’s unclear how the nonlinear coupling strength κ in
Eq. (6.1) will need to scale with N for any particular QC application. Our results are
nonetheless compelling because as N increases, the new interactions generated have the
same squeezing strength as existing ones—i.e., existing squeezing is not “redistributed” to
the new pairings as the lattice and pump power grow. Moreover, scalability does enable
quantum encoding redundancy.
While finite squeezing errors can be mitigated in medium-sized proof-of-principle experiments with CV cluster states (Chapter 3), and some work also addresses CV error correction
in general [56, 57, 59]. Also, as mentioned at the end of Chapter 2, given the ubiquity of
experimental errors, the bandwidth of any continuous-variable quantum computer is most
likely finite, which motivates the study of encoding a lower-dimensional system (e.g., a
qubit) into each oscillator or field mode [57, 61]. This would allow for “continuous-variable
quantum hardware” to be used for standard qubit algorithms and suggests the possibly
that fault tolerance could be addressed within the qubit framework [62], but it is unknown
whether this model is compatible with the one-way QC model presented here. Thus, fault
tolerance in the continuous-variable case remains an important open question—and especially for CV cluster-state QC. We hope to spur further investigations along these lines.
6.7
Conclusion
We have presented a theoretical breakthrough that opens the door to large-scale experimental generation of a universal one-way quantum computing resource, using existing technology. The entangled states produced by this method will also be objects of interest in the
73
study of entanglement at mesoscopic scales. Further simplification of the method may be
possible using additional degrees of freedom, such as spatial modes.
This method of entangling—in one fell swoop—a large optical frequency comb into a
continuous-variable cluster state is the first “top-down” approach proposed for one-way
quantum computation using optical encodings of information in either discrete or continuous variables. While open questions remain about the effects of finite squeezing on
scalability for particular quantum computing tasks, the unprecedented scalability of this
method encourages further theoretical research. Experimental implementation is already
underway.
74
Part II
Trapped Ion Physics
75
Chapter 7
Trapping and Controlling Ions
Trapping ions with electromagnetic fields is a technology that is 30 years in the making.
Recent improvements in spectrally narrow coherent sources of laser light now allow these
isolated quantum systems to be precisely controlled. While the experimental science involved is rather complex, the maturity of the technology means that comprehensive reviews
are available at varying levels of complexity. This chapter uses Chapter 17 of Walls and
Milburn [54], which presents a brief overview of ion trap technology, as well as the comprehensive review by Leibfried et al. [101], focussing on quantum dynamics, and the article by
James [102], describing the trapping of multiple ions. With so much material available, the
focus here will be limited to the principles needed to understand my research described in
the next two chapters. The reader is encouraged to consult the resources mentioned here
and the references they contain for a more in-depth and comprehensive analysis.
7.1
Trapping a Single Ion
In order to use a single ion as a controlled independent quantum system, we first have
to isolate (trap) it. Laplace’s equation dictates that electrostatic fields are insufficient for
this purpose since the potential always contains one unstable (untrapped) direction. Timedependent potentials are therefore needed. All of the work considered in this thesis has in
mind the model of a linear ion trap shown in Figure 7.1.
The time-dependent potential felt by the ion at the center of the trap has the approximate form
Φ(x, y, z, t) '
V̄
V0
(kx x2 + ky y 2 + kz z 2 ) +
cos(ωrf t)(kx0 x2 + ky0 y 2 ) ,
2
2
76
(7.1)
y
x
z
y
x
RF
+ DC
z
– DC
Figure 7.1: Schematic of a linear ion trap, based on Ref. [101]. The static positive (dark blue)
and negative (light blue) electrodes generate a potential well in the z-direction (trap axis) for
the positively charged ion (red ball), while a common oscillating (radio frequency) potential
is generated by the two green electrodes, confining the ion dynamically in the x-y plane. The
inset at the top right shows an end-on view down the trap axis. The checkered pattern of light
and dark blue labels the static electrodes’ position in the x-y plane.
where the first group of terms corresponds to an overall static potential, while the second
refers to the (ωrf -periodic) time-dependent portion. Laplace’s equation requires
kx + ky + kz = 0
kx0 + ky0 = 0 .
and
(7.2)
(7.3)
With foresight, the following unitless parameters may be defined for each j ∈ {x, y, z}:
4Z |e| V̄ kj
,
2
mωrf
2Z |e| V 0 kj0
,
qj :=
2
mωrf
aj :=
(7.4)
(7.5)
where m is the ion’s mass and Z |e| its charge (with kz0 = 0). If ωrf is large enough, then
aj , qj 1, and we can model the motion of the ion as consisting of harmonic motion at the
secular frequency
νj =
ωrf q
aj + qj2 ,
2
(7.6)
with small-amplitude harmonic micromotion at the frequency ωrf superimposed on top of
it. In most cases of experimental interest the micromotion can be neglected [101], letting
us model the motion of the ion harmonically:
H = ~νx a†x ax + ~νy a†y ay + ~νz a†z az .
|
{z
} | {z }
transverse
77
longitudinal
(7.7)
Since the design of Figure 7.1 ensures that kz0 = 0, then also qz = 0, and thus
r
νz =
Z |e| V̄ kz
.
m
(7.8)
Typically, transverse trapping frequencies νx and νy are much higher than νz , meaning the
motion is approximately one-dimensional. In this case, we can neglect transverse motion
altogether (which is, in any case, decoupled from longitudinal oscillations), in addition
to neglecting the micromotion, resulting in a simple one-dimensional harmonic potential
experienced by the ion in the z-direction:
H = ~ν a† a ,
(7.9)
where ν = νz , and we can drop the z-subscripts on the operators, as well. Under this
Hamiltonian, the position and momentum operators for the ion are given, respectively, by1
r
q=
~
(a + a† )
2mν
(7.10)
and
r
p = −i
7.2
~mν
(a − a† ) .
2
(7.11)
Laser Coupling and Cooling
With our ion trapped and effectively confined to one-dimensional motion, it still remains
in a highly excited vibrational state corresponding to temperatures ∼ 104 K. In order to
access the quantum dynamics of the ion, we first need to cool it. This usually proceeds in
two stages [101].
The first is Doppler cooling, whereby a laser is detuned below the resonance frequency
of a short-lifetime atomic transition and trained on the ion along the trap axis. Using
a semiclassical picture, in the ion’s reference frame, the laser photons become blueshifted
closer to resonance for states of momentum opposite to the photons’ wavevector k, while
becoming redshifted further off resonance for states of momentum in the same direction
as k. This asymmetry causes the ion to absorb a laser photon preferentially when it is
counterpropagating against the laser beam. These absorbed photons transfer momentum to
the ion preferentially against its direction of propagation, while the momentum distribution
of the spontaneously emitted photons has no such preference, resulting in a net loss in
1
Phase and prefactor conventions vary in the literature.
78
kinetic energy for the ion on average. While this method is efficient and effective as a firststage cooling procedure, reliable preparation of the vibrational ground state is not possible
due to heating from off-resonance absorption [101].
The second stage is known as resolved sideband cooling, which relies on the ability to
detune selectively below a metastable transition—that is, a transition whose spontaneous
emission rate Γ ν, where ν is the (z-axis, secular) trap frequency. The long-lived nature
of the state is necessary in order to narrow the transition linewidth so that an external
laser can be tuned to a frequency within ν of the atomic transition frequency ωA without
simultaneously exciting the transition directly; that is, ωL = ωA − ν. The detuning by one
unit of trap frequency and the coupling of vibrational motion and electronic states (described
below) allows for the absorption of one laser photon at frequency ωL and one vibrational
phonon at frequency ν to excite the meta-stable transition. While Doppler cooling relies
on short-lifetime (usually, dipole) transitions in order to ensure broad absorption and rapid
dissipation, the ones desired for resolved sideband cooling are often quadrupole transitions
or Raman (stimulated two-photon) transitions. Using this technique, the vibrational ground
state may be prepared with a probability greater than 99% [101].
Fortunately, the formalism for all three types of transitions—including Raman transitions, which use a stimulated transition to an auxiliary level—can be modeled using the
two-state formalism for qubits [101] discussed in Chapter 2. The correspondence is given
by |gi ↔ |1i and |ei ↔ |0i, where |gi and |ei are the ground state and desired excited state
of the atom, respectively. Using this correspondence, the Hamiltonian describing the atom
interacting with a classical laser field can be written2
h
i
H = ~ν a† a + ~ωA σz − i~Ω0 σ+ e−i(ωL t−k cos θ q) − σ− ei(ωL t−k cos θ q) ,
| {z } | {z } |
{z
}
vibrational electronic
(7.12)
laser-ion interaction
where Ω0 is the Rabi frequency for the laser-ion interaction, k is the magnitude of the
wavevector k, which makes an angle θ with the trap axis, σz = |eihe| − |gihg| is a Pauli
matrix, σ+ = |eihg|, and σ− = |gihe|. The crucial point is that the phase of the laser field
as seen by the ion depends on the position of the ion, allowing coupling of the vibrational
motion (through q) to an atomic transition (through σ± ).
In the interaction picture generated from the first two terms in Eq. (7.12), the interaction
Hamiltonian (third term) becomes
h
i
HI (t) = −i~Ω0 σ+ (t)eik cos θ q(t) − σ− (t)e−ik cos θ q(t) ,
(7.13)
where the interaction-picture electronic raising and lowering operators are
σ± (t) = e±i∆t σ± ,
2
(7.14)
Conventions vary regarding the phase of the interaction terms and factors of two in the Rabi frequency.
The Hamiltonian in Eq. (7.12) is consistent with that used in the following two chapters.
79
the interaction-picture position operator for the ion is
r
q(t) =
~
(ae−iνt + a† eiνt ) ,
2mν
(7.15)
and the detuning of the laser below the atomic transition is given by
∆ = ωA − ωL .
(7.16)
The size of the rms fluctuation in q as compared to the wavelength of the laser is measured
by the Lamb-Dicke parameter
r
η :=
~k 2 cos2 θ
∆xrms
=
(2π cos θ) .
2mν
λL
(7.17)
Typical values of the Lamb-Dicke parameter are η ∼ 0.01 to 0.1 [54]. When η 1, the
so-called “Lamb-Dicke limit,” the ion is well localized with respect to the wavelength of
the laser, and we can expand the exponentials in Eq. (7.13) to first order in η, which is
equivalent to first order in k cos θ q(t), giving
h
i
HI (t) ' −i~Ω0 σ+ (t) 1 + ik cos θ q(t) − σ− (t) 1 − ik cos θ q(t)
h
i
h
i
= ~Ω0 −iσ+ (t) + iσ− (t) + ~Ω0 k cos θ q(t) σ+ (t) + σ− (t)
= ~Ω0 σy (t) + ~Ω0 k cos θ q(t)σx (t) ,
{z
}
| {z } |
carrier
(7.18)
sideband
where σx (t) = σ+ (t)+σ− (t) and σy (t) = −iσ+ (t)+iσ− (t), along with σz (t) above, complete
the set of interaction-picture Pauli matrices for the atomic transition. The first term corresponds to excitation of the transition directly by the laser, while the second couples the
atomic transition to vibrational motion. In order to see why these are labeled as “carrier”
and “sideband,” it helps to write out the two explicitly:
HI (t) = ~Ω0 (−iσ+ e+i∆t + iσ− e+i∆t ) + ~Ω0 η(ae−iνt + a† eiνt )(σ+ e+i∆t + σ− e−i∆t ) .
|
{z
} |
{z
}
carrier
sideband
(7.19)
The particular transitions excited depend on the detuning ∆. We will examine three
cases: ∆ ∈ {0, ν, −ν}, corresponding to carrier, first red sideband, and first blue sideband
excitation, respectively.
The carrier transition, ∆ = 0, corresponds to direct excitation of the atomic transition.
The only resonant (non-oscillatory) terms in Eq. (7.19) in this case come from the “carrier”
80
term:
carrier
HI (t) −−−−−→ −i~Ω0 (σ+ − σ− ) ,
(7.20)
(∆ = 0)
which directly excites transitions between the |gi and |ei electronic states but does not
couple to the vibrational state at all. The first red sideband transition is obtained by
setting ∆ = ωA − ωL = ν; that is, the laser is detuned one unit of vibrational energy below
(to the red of) the atomic transition. Resonant terms in this case come from the “sideband”
term only:
red sideband
HI (t) −−−−−−−−→ ~Ω0 η(aσ+ + a† σ− ) ,
(7.21)
(∆ = ν)
which is of Jaynes-Cummings form [101, 103], corresponding to excitation of the atomic
transition with energy gap ~ωA = ~ωL + ~ν upon absorption of one laser photon at energy ~ωL , along with absorption of one vibrational phonon at energy ~ν. Detuning the laser
above (to the blue of) the atomic transition by one unit of vibrational energy (∆ = −ν)
generates the blue sideband transition, which has resonant terms
blue sideband
HI (t) −−−−−−−−→ ~Ω0 η(a† σ+ + aσ− ) ,
(∆ = −ν)
(7.22)
corresponding to atomic excitation of the transition at energy ~ωA = ~ωL − ~ν upon absorption of one laser photon at energy ~ωL , along with emission of one vibrational phonon
at energy ~ν. These three cases are illustrated in Figure 7.2.
The carrier transition can be used to modify the electronic state independent of the
vibrational motion. It thus provides no coupling between the two degrees of freedom. The
sideband transitions, on the other hand, couple electronic transitions to vibrational motion.
This has two uses that will be discussed here. The first is resolved sideband cooling, as
mentioned earlier in this section. Detuning the laser to the red sideband generates an
interaction of the form of (7.21), which selectively removes one quantum of vibration per
cooling cycle, exciting the atom in the process. Since the electronic transition is longlived, it can be optically pumped back down to the ground state to complete the cycle.
In the Lamb-Dicke limit, the emitted photon will almost always be emitted on the carrier
transition, resulting in no recoil [104].
The second use of sideband transitions is as a motional detector. Originally envisioned
for a continuous scalar field, a simple (and tunable) De Witt monopole coupling [105] takes
the form of the “sideband” term in Eq. (7.18):
HDWM ∼ q(t)σx (t) ,
81
(7.23)
Figure 7.2: Laser-induced electronic transitions in a trapped ion. The carrier transition (left)
corresponds to absorption of one laser photon tuned to the resonance frequency of the atom;
no coupling to the vibrational motion is involved. The red sideband transition (center) allows
the atom to act as its own motional detector, becoming excited at the absorption of one laser
photon plus one vibrational phonon. The blue sideband transition (right) involves absorption
of one laser photon and the emission of one phonon in order to excite the atom.
where the time-dependent position operator q(t) acts like a discretized scalar field (of
phonons), and the resonant frequency of the detector is ∆, which is tunable by changing the laser frequency and is embedded in the definition σx (t) = (σ+ e+i∆t + σ− e−i∆t ). In
this way, the ion acts as its own motional detector via red sideband-detuned coupling by an
external laser. While Eq. (7.23) suffices to couple electronic and vibrational motion, if it
is to be useful as a detection mechanism, we must be able to read out the electronic state.
This is accomplished by a state-dependent fluorescent shelving scheme [101]: a strong probe
laser causes ions in the excited electronic state |ei to fluoresce, while those in the ground
electronic state |gi do not, corresponding with high efficiency to a projective measurement
in the basis {|gi , |ei}. This process is illustrated in Figure 7.3. The monopole coupling
in Eq. (7.23), along with this method of readout, will be used extensively in the next two
chapters as the main mechanism for detecting vibrations in the ion.
7.3
Multiple Ions
These methods can easily be extended to more than one ion and their collective normal
modes of vibration [102]. If axial trapping potential is kept shallow enough, multiple ions
can be trapped in a line along the trap axis, up to about 10 or so. Beyond this number,
the one-dimensional approximation breaks down as the ions find energetically favorable
zig-zig configurations. Here I derive the dynamics for multiple ions in the trap under the
one-dimensional assumption, following Ref. [102] closely.
82
probe
Figure 7.3: Measurement of an ion’s electronic state by fluorescent shelving. A strong probe
laser excites a short-lived transition from the excited state |ei to an auxilliary state |ai but
does not couple to the ground state |gi at all. If the ion is excited, the probe laser repeatedly
excites the atom further into the auxilliary state, which then quickly decays, emitting many
photons in all directions. This fluorescent signal is easily detected and corresponds with high
efficiency to projective measurement in the basis {|gi , |ei}.
The Lagrangian for the ions in the trap is
L=
m T
ẋ ẋ − V (x) ,
2
(7.24)
where x = (x1 , . . . , xN )T is a column-vector containing the position of each ion relative to an
arbitrary, fixed zero along the trap axis, and the dot denotes time derivative. The potential
V (x) is composed of the one-dimensional effective harmonic potential and a Coulomb term
coming from the mutual repulsion of the ions:
Z 2 e2
|xj − xk |−1
4π0
{j,k|j>k}
X
m 2 T
m
= ν x x + ν 2 `3
|xj − xk |−1 ,
2
2
V (x) =
m 2 T
ν x x+
2
X
(7.25)
{j,k|j6=k}
where the length scale ` has been defined via
`3 :=
Z 2 e2
.
4π0 mν 2
83
(7.26)
N = 10
N=9
N=8
N=7
N=6
N=5
N=4
N=3
N=2
N=1
symmetry axis
Figure 7.4: Equilibrium positions for multiple ions in a linear trap, using data from Ref. [102].
The equilibrium position for each ion results from balancing the external trapping force (harmonic potential) with the Coulomb repulsion from all the other ions, Eq. (7.28). Small oscillations about these points can be approximated by a linear system of coupled harmonic
oscillators. Notice that, especially for larger N , ions on the ends are further apart than those
in the middle of the trap. The scale factor ` is defined in Eq. (7.26).
We now define equilibrium positions x0 for the ions via
∂
0=
V ∂xm x=x0
= mν 2 x0m + mν 2 `3
X
(x0j − x0m )−2 − mν 2 `3
j>m
(x0j − x0m )−2
j<m


= mν 2 ` um +
X
X
(uj − um )−2 −
j>m
X
(uj − um )−2  ,
(7.27)
j<m
where u := x0 /` is constant and satisfies
0 = um +
X
(uj − um )−2 −
j>m
X
(uj − um )−2 .
(7.28)
j<m
The equilibrium positions for multiple ions (up to N = 10) are plotted in Figure 7.4, along
with the scale factor `. Notice that the scaled equilibrium positions u don’t depend on the
physical parameters of the trap or of the ions, all of which is consolidated into the scale
factor `—a remarkable property of the combination of Coulomb and harmonic potentials.
84
Defining the vector of displacements from the equilibrium q(t) = x(t) − x0 , we can
expand the potential to quadratic order in q:
1
V (x) ' V (x0 ) + qT Wq ,
2
(7.29)
∂2
:=
V .
∂xm ∂xn x=x0
(7.30)
where the entries in W are
Wmn
To evaluate this matrix, let’s start with the off-diagonal elements first. Thus, assume that
m 6= n. Then,

(m 6= n)
Wmn = mν 2 `3
∂ 
∂xn

X
(xj − xm )−2 −
j>m
X
(xj − xm )−2 
j<m
x=x0
−3
= mν 2 `3 (−2)x0n − x0m −3
= mν 2 (−2)un − um .
(7.31)
The diagonal terms are given by


∂  −3
` xm +
(xj − xm )−2 −
(xj − xm )−2 
∂xm
j>m
j<m
x=x0


X
= mν 2 `3 `−3 + 2
|x0j − x0m |−3 
X
Wmm = mν 2 `3
X
j6=m


= mν 2 1 + 2
X
|uj − um |−3  .
(7.32)
j6=m
Once again, all dependence on physical parameters can be factored out, allowing us to define
a numerical matrix A satisfying
W = mν 2 A ,
(7.33)
Plugging this into Eq. (7.29) and the result into Eq. (7.24) gives
m T
m
ẋ ẋ − V (x0 ) − ν 2 qT Aq
2
2
m 2 T
m T
→ ẋ ẋ − ν q Aq
2
2
m T
m 2 T
= q̇ q̇ − ν q Aq ,
2
2
L'
85
(7.34)
where in the second line we have ignored the overall change in energy that is independent
of the dynamical variables. This is the Lagrangian for a linear system of coupled harmonic
oscillators. The eigendecomposition of A = BT ΛB, where Λ = diag(µ1 , . . . , µN ), arranged
in ascending order, allows Eq. (7.34) to be rewritten as
m T T
m
q̇ (B B)q̇ − ν 2 qT (BT ΛB)q
2
2
m 2
m
T
= (Bq̇) (Bq̇) − ν (Bq)T Λ(Bq)
2
2
m T
m 2 T
= Q̇ Q̇ − ν Q ΛQ
2
2
mX 2 mX 2 2
Q̇ −
ν Q ,
=
2 p p
2 p p p
L=
(7.35)
where
Qp :=
X
b(p)
m qm
(7.36)
m
(p)
are normal mode coordinates (bm are the entries of B), and
√
νp := ν µp
(7.37)
is the frequency of normal mode p. The first normal mode (numbered in order of increasing
frequency) is called the center of mass mode and corresponds to all ions moving together
as a single unit as if they were clamped together rigidly, while the second is the breathing
mode, in which the displacement of each ion is proportional to the distance of the ion from
the trap center. The number of normal modes always equals the number of ions, with higher
modes corresponding to more complex coherent motion.
The normal modes oscillate independently, so quantization results in a simple Hamiltonian:
H=
X
~νp a†p ap ,
(7.38)
p
where ap and a†p are raising and lowering operators for mode p. The quantized position and
momentum operators for the normal modes (also known as the global modes) are defined
by
s
Qp =
~
(ap + a†p )
2mνp
86
(7.39)
and
r
Pp = −i
while those for the local modes are
r
qm =
~mνp
(ap − a†p ) ,
2
(p)
~ X bm
(ap + a†p )
2mν p µ1/4
p
(7.40)
(7.41)
and
r
pm = −i
~mν X 1/4 (p)
µp bm (ap − a†p ) .
2
p
(7.42)
This linear harmonic system behaves approximately as a discretized (and nonuniform) scalar
field in which qm plays the role of the local phonon field operator at the equilibrium position x0m . This picture is explored further in Chapter 9.
7.4
Applications
Most interest in ion trap technology from the quantum information community comes from
its potential for use as a quantum computing platform [104, 106, 107, 108]. While much
progress has been made in this direction, my work focuses instead on trapped ions as
an extremely well-controled quantum system, with the ability to prepare many types of
quantum states with high fidelity [101, 109] and realize precise quantum oeprations [110,
111]. Quantum information theory is not only about the processing of classical information
using quantum means (e.g., Shor’s factoring algorithm [15]) but also includes the ability of
one quantum system to efficiently simulate another [26]. The control afforded to ion trap
technology makes it a prime candidate for this purpose. My work follows along these lines.
Chapter 8 proposes using a single trapped ion to simulate the dynamics of a generic
quantum harmonic oscillator with time-dependent frequency [5]. This is accomplished by
modulating the trap frequency in real time by varying the endcap voltage (dark blue electrodes in Figure 7.1) at a rate low enough to still allow the harmonic approximation to be
valid. Chapter 9 examines correlations in multiple ions in a static trap as a function of the
initial state [6]. Both proposals rely on the ability to prepare a variety of interesting initial
states with high fidelity [101] and to make localized detection of vibrational motion using
the laser coupling discussed at the end of Section 7.2. We turn now to the use of a single
trapped ion as a time-dependent harmonic oscillator.
87
Chapter 8
A Single Trapped Ion as a
Time-Dependent Harmonic
Oscillator
This chapter closely follows Ref. [5] in a self-contained fashion.
8.1
Introduction
The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic time-dependent Hamiltonian systems and has been applied to a wide range of
physical problems by choosing the mass, the frequency, or both, to be time-dependent. The
earliest application is to squeezed state generation in quantum optics [112, 113, 114], in
which the effect of a second-order optical nonlinearity on a single-mode field can be modeled by a harmonic oscillator with a frequency that is harmonically modulated at twice the
bare oscillator frequency. It was subsequently shown that any modulation of the frequency
could produce squeezing [115], and thus the same model could be used to approximately
describe the generation of photons in a cavity with a time-dependent boundary [116, 117].
The model has been used in a number of quantum cosmological models. In Ref. [118],
a time-dependent frequency has been used to explain entropy production in a quantum
mini-superspace model. The model, with both mass and frequency time-dependent, has
been particularly important in developing an understanding of how quantum fluctuations
in a scalar field can drive classical metric fluctuation during inflation [119, 120]. In a
cosmological setting the time-dependence is not harmonic and is usually exponential. In
all physical applications, of course, the model is only an approximation to the true physics,
and its validity can be tested only with considerable difficulty, especially in the cosmological
setting. Here we propose a realistic experimental context in which the time-dependent
quantum harmonic oscillator can be studied directly.
88
Many decades of effort to refine spectroscopic measurements for time standards now
enable a single ion to be confined in three dimensions, its vibrational motion restricted
effectively to one dimension, and the ion cooled to the vibrational ground state with a
probability greater than 99% [101]. Laser cooling is based on the ability to couple an internal
electronic transition to the vibrational motion of the ion [121]. These methods can easily be
extended to more than one ion and their collective normal modes of vibration [102]. Indeed
so carefully can the coupling between the electronic and vibrational states be engineered
that is is possible to realise simple quantum information processing tasks [110, 111]. We
use the control of trapping potential afforded by ion traps, together with the ability to
reach quantum limited motion, to propose a simple experimental test of quantum harmonic
oscillators with time-dependent frequencies. We also make use of the ability to make highly
efficient quantum measurements, based on fluorescent shelving [101], to propose a practical
means to test our predictions.
In this chapter, we calculate the excitation probability of a trapped ion in a general timedependent potential. When beginning in the vibrational ground state of the unchirped trap
and starting the chirping process adiabatically, the excitation probability is simply related
to the Fourier transform of the solution of the Heisenberg equations of motion (which is
also the same as the trajectory of the equivalent classical oscillator). We compare our result
with that of Ref. [122] for the case of a single ion undergoing an exponential frequency
chirp. The cited work attempts to use this experimental setup to model a massless scalar
field during an inflating (i.e., de Sitter) universe, which would give a thermal excitation
spectrum as a function of the detector response frequency [123]. The analysis is incorrect,
however, because the wrong Hamiltonian was used. Nevertheless, the corrected calculation
presented here also gives an excitation spectrum with a thermal signature, although the
particular functional form is different.
8.2
General Solution
The quantum Hamiltonian for a single ion in a time-dependent harmonic trap can be wellapproximated in one dimension by
H=
M
p2
+
ν(t)2 q 2 ,
2M
2
(8.1)
where ν(t) is time-dependent but always assumed to be much slower than the timescale of
the micromotion [101]. For emphasis, we have indicated the explicit time-dependence of
the frequency ν; we will often omit this from now on. Working in the Heisenberg picture,
89
we get the following equations of motion for q and p:
p
,
M
ṗ = −M ν 2 q .
q̇ =
(8.2)
(8.3)
Dots indicate total derivatives with respect to time. Differentiating again and plugging in
these results gives
0 = q̈ + ν 2 q ,
ν̇
0 = p̈ − 2 ṗ + ν 2 p .
ν
(8.4)
(8.5)
As we shall see, only Eq. (8.4) is necessary for calculating excitation probabilities, so we
will focus only on it. These equations are operator equations, but they are identical to the
classical equations of motion for the analogous classical system. Interpreting them as such,
we will label the two linearly independent c-number solutions as h(t) and g(t), where the
following initial conditions are satisfied:
h(0) = ġ(0) = 1
and
ḣ(0) = g(0) = 0 ,
(8.6)
Writing q(0) = q0 and p(0) = p0 , the unique solution for q to the initial value problem above
is
q(t) = q0 h(t) +
p0
g(t) .
M
(8.7)
By differentiating and using the relations above, we know also that
p(t) = M q0 ḣ(t) + p0 ġ(t) .
(8.8)
To check our math, we can verify that [q(t), p(t)] = i~, which is fulfilled if and only if the
Wronskian W (h, g) of the two solutions is one for all times—specifically,
hġ − ḣg = 1 ,
(8.9)
where we have assumed that [q0 , p0 ] = i~.
Moreover, if the initial state at t = 0 is symmetric with respect to phase-space rotations,
then we have additional rotational freedom in choosing the initial quadratures. (This would
be the case, for instance, if we start in the instantaneous ground state.) Notice that Eq. (8.7)
can be written as the inner product of two vectors:
p0 q(t) = q0 ,
· h(t), ν0 g(t)
M ν0
90
(8.10)
(and similarly for Eq. (8.8)), where we have normalized the quadrature operators to have the
same units. As an inner product, this expression is invariant under simultaneous rotations
of both vectors. Thus, if the initial state possesses rotational symmetry in the phase plane,
then the rotated quadratures are equally as valid as the original ones for representing the
initial state, which means that an arbitrary rotation can be applied to the second vector
above without changing any measurable property of the system. This freedom can be used,
for instance, to define new functions h0 (t) and g 0 (t) that are more convenient for calculations,
where the linear transformation between them and the original ones (with prefactors as in
Eq. (8.10)) is a rotation. We will use this freedom in the next section.
One reason why ion traps have become a leading implementation for quantum information processing is the ability to efficiently read out the internal electronic state using a
fluorescence shelving scheme [101]. As the internal state can become correlated with the
vibrational motion of the ion, this scheme can be configured as a way to measure the vibrational state directly [124]. To correlate the internal electronic state with the motion of
the ion, an external laser can be used to drive an electronic transition between two levels |gi and |ei, separated in energy by ~ωA . The interaction between an external classical
laser field and the ion is described, in the dipole and rotating-wave approximation, by the
interaction-picture Hamiltonian [101]
h
ik cos θq(t)
HL = −i~Ω0 σ+ (t)e
−ik cos θq(t)
− σ− (t)e
i
,
(8.11)
where Ω0 is the Rabi frequency for the laser-atom interaction, ωL is the laser frequency, k
is the magnitude of the wave vector k, which makes an angle θ with the trap axis, q(t) is
given in Eq. (8.7), and
σ± (t) = e±i∆t σ± .
(8.12)
The electronic-state raising and lowering operators are defined as σ+ = |eihg| and σ− =
|gihe|, respectively, and
∆ = ωA − ωL
(8.13)
is the detuning of the laser below the atomic transition. We can construct a meaningful
quantity that characterizes the “size” of q(t) based on the width of the ground-state wave
p
packet for an oscillator with frequency ν(t), namely ~/2M ν(t). As long as this quantity
is much smaller than k cos θ throughout the chirping process, then we can expand the
exponentials in Eq. (8.11) to first order and define the interaction Hamiltonian HI between
the electronic states and vibrational motion (still in the interaction picture) by
HI = ~Ω0 k cos θq(t) e−i∆t σ− + e+i∆t σ+ .
(8.14)
where we have assumed that ωL is far off-resonance, and thus ∆ 6' 0. However, we will not
make the rotating wave approximation here because ν is time-dependent.
91
Using first-order time-dependent perturbation theory, the probability to find the ion in
the excited state is
P
(1)
1
= 2
~
Z
T
Z
T
dt2 hHI (t1 )Pe HI (t2 )i
Z
2 2
2
= Ω0 k cos θ
dt1
0
0
T
Z
T
dt1
0
dt2 e−i∆(t1 −t2 ) hq(t1 )q(t2 )i , (8.15)
0
where Pe = 1vib ⊗ |eihe| is the projector onto the excited electronic state (and the identity
on the vibrational subspace). We always assume that the ion begins in the electronic ground
state. If the ion also starts out in the instantaneous vibrational ground state for a static
trap of frequency ν0 = ν(0) at t = 0 (which is most useful when the chirping begins in the
adiabatic regime), then we can evaluate the two-time correlation function as
2
2
p
hq(t1 )q(t2 )iground = q0 h(t1 )h(t2 ) + 02 g(t1 )g(t2 )
M
hq0 p0 i h(t1 )g(t2 ) − h(t2 )g(t1 )
+
M
ih
i
~ h
=
h(t1 ) − iν0 g(t1 ) h(t2 ) + iν0 g(t2 )
2M ν0
~
=
f (t1 )f ∗ (t2 ) ,
(8.16)
2M ν0
where we have used the facts that for the vibrational ground state, q02 = (p0 /M ν0 )2 =
~/2M ν0 and hq0 p0 i =
1
2
h{q0 , p0 } + [q0 , p0 ]i = i~/2, and we have defined the complex func-
tion
f (t) = h(t) − iν0 g(t) ,
(8.17)
which is the solution to Eq. (8.4) with initial the conditions, f (0) = 1 and f˙(0) = −iν0 .
Plugging this into Eq. (8.15) gives, quite simply,
P (1) → (Ω0 η0 )2 |F|2 ,
(8.18)
where
Z
F=
T
dt e−i∆t f (t) ,
(8.19)
0
and we have defined the unitless, time-dependent Lamb-Dicke parameter [101] as
s
η(t) =
~k 2 cos2 θ
,
2M ν(t)
92
(8.20)
and η0 = η(0). Recalling that f (t) can be considered a complex c-number solution to the
equations of motion for the equivalent classical Hamiltonian, Eq. (8.18) shows that the
excitation probability is simply related to the Fourier transform of the classical trajectories
when beginning in the vibrational ground state.
8.3
Exponential Chirping
Recent work [122] has suggested that an exponentially decaying trap frequency has the same
effect on the phonon modes of a string of ions as an expanding (i.e., de Sitter) spacetime
does on a one-dimensional scalar field [125]. An inertial detector that responds to such an
expanding scalar field would register a thermal bath of particles, called Gibbons-Hawking
radiation [123]. Ref. [122] suggests that the acoustic analog [126] of this radiation could be
seen in an ion trap, causing each ion to be excited with a thermal spectrum with temperature
~κ/2πkB , as a function of the detuning ∆, where κ is the trap-frequency decay rate. The
analysis used an incorrect Hamiltonian that neglected squeezing and source terms that
have no analog in the expanding scalar field model but which are present when considering
trapped ions in this way, and the results are incorrect. In this section, we revisit this
problem and calculate the excitation probability for a single ion in an exponentially decaying
harmonic potential, as a function of the detuning ∆.
We write the time-dependent frequency as
1
ν(t) = ν0 e−κt .
(8.21)
q̈ + ν02 e−2κt q = 0 .
(8.22)
This results in
Solutions with initial conditions (8.6) are
ν ν i
πν0 h ν0 ν 0
J1
Y0
− Y1
J0
,
2κ h
κ
κ
κ
κ i
π
ν0
ν
ν0
ν
−J0
Y0
+ Y0
J0
,
g(t) =
2κ
κ
κ
κ
κ
h(t) =
(8.23)
(8.24)
where the time dependence is carried in ν = ν(t) from Eq. (8.21), and Jn and Yn are Bessel
functions. We could plug these directly into the formulas from the last section, but we will
simplify the calculations by considering the limits of slow and long-time frequency decay,
represented by
ν0 κ
ν0 e−κT κ ,
and
1
(8.25)
The authors of Ref. [122] consider both signs in the exponential, but we will restrict ourselves to the
case that allows us to begin chirping in the adiabatic limit.
93
respectively. This allows us to do several things. First, it allows us to use the usual ground
state of the unchirped trap at frequency ν0 as a good approximation to the ground state of
the expanding trap at t = 0, since at that time the system is being chirped adiabatically.
This is important because it allows the experiment to begin with a static potential, which
is useful for cooling. Second, it allows us to simplify h(t) and g(t) using the phase-space
rotation freedom discussed above. Using asymptotic approximations for the Bessel functions
in the coefficients,
J0
ν 0
r
ν
π
2κ
0
cos
−
,
κ
πν0
κ
4
ν r 2κ
ν
π
0
0
Y0
'
sin
−
,
κ
πν0
κ
4
' −Y1
κ
ν 0
'
J1
κ
ν 0
'
(8.26)
(8.27)
we get
r
ν ν i
πν0 h
sin ϕ Y0
+ cos ϕ J0
,
2κ
κ
κ
r
ν ν i
πν0 h
− cos ϕ Y0
+ sin ϕ J0
.
ν0 g(t) '
2κ
κ
κ
h(t) '
(8.28)
(8.29)
where ϕ = ν0 /κ − π/4. Since we are taking the initial state to be the ground state, which is
symmetric with respect to phase-space rotations, we can use the freedom discussed in the
previous section to undo the rotation represented by Eqs. (8.28) and (8.29) and define the
simpler functions
r
πν0 ν ,
Y0
2κ
κ
r
ν π
g(t) → g 0 (t) =
J0
.
2κν0
κ
0
h(t) → h (t) =
(8.30)
(8.31)
The primes are unnecessary due to the symmetry of the initial state, so we drop them from
now on and plug directly into Eq. (8.17):
r
ν i
πν0 h ν Y0
− iJ0
2κ
κ
κ
r
πν0 (1) ν
= −i
H
,
2κ 0
κ
f (t) =
(8.32)
(1)
where Hn is a Hankel function of the first kind. The integral in Eq. (8.19) can be evaluated
in the limits (8.25) using techniques similar to those used in Ref. [122]. First, define
eα =
ν
,
κ
τ = α − κt ,
u = eτ ,
94
and x = ∆/κ .
(8.33)
The integral in question then becomes (neglecting the prefactor)
T
Z
−i∆t
dt e
0
(1)
H0
1
=
κ
Z
ν κ
(1)
dt e−i∆t H0 (eα−κt )
(1)
α−κT
Z
e−ixα
Z
κ
T
=
0
α
e−ixα
→
κ
=
Z
dτ e−ix(α−τ ) H0 (eτ )
∞
−∞
∞
0
(1)
dτ eixτ H0 (eτ )
(1)
du uix−1 H0 (u) .
(8.34)
Inserting a convergence factor with x → x − i, and then taking the limit → 0+ , we can
use the formula
Z
0
∞
(1)
du uix−1 H0 (u) = −2ix
Γ(ix/2)
(eπx − 1)Γ(1 − ix/2)
(8.35)
to evaluate
πν0 1 Γ(ix/2) 2
1
|F| =
2κ κ2 Γ(1 − ix/2) (eπx − 1)2
2πν0
1
= 3 2 πx
.
κ x (e − 1)2
2
(8.36)
When plugging in for the dummy variables (8.33), this gives
P (1) = (Ω0 η0 )2
2πν0
1
.
κ∆2 (eπ∆/κ − 1)2
(8.37)
The calculated result from Ref. [122] for a single ion is
(1)
PGH = (Ω0 η0 )2
2π
1
,
κ∆ e2π∆/κ − 1
(8.38)
which contains a Planck factor with Gibbons-Hawking [123] temperature T = ~κ/2πkB but
is different from the actual result for a single ion, given by Eq. (8.37).
Several things should be noted about these functions. First, they both break down
as ∆ → 0 because of the approximation made in obtaining Eq. (8.14). They also fail if
the time-dependent Lamb-Dicke parameter (8.20) ever becomes too large throughout the
chirping process. Furthermore, most cases of interest will be ∆ ' ν0 (the first red sideband)
and near ∆ ' −ν0 (the first blue sideband), which means that |∆| κ, since ν0 κ. The
first red sideband represents a detector that requires the absorption of one phonon (plus one
laser photon) in order to excite the atom—the usual thing we mean by “particle detector”
when the particles are phonons. The first blue sideband, on the other hand, represents a
95
detector that emits a phonon in order to excite the atom (along with absorbing one laser
photon).
There are a couple of ways to compare these functions. First, we can take the ratio of
the two for both the red- and blue-sideband cases. In both cases, we obtain
P (1)
(1)
PGH
'
ν0
(1 + 2e−π|∆|/κ )
|∆|
(8.39)
plus terms of order O(e−2π|∆|/κ ). Since |∆| ' ν0 , the prefactor is close to one, and the second
term is very small (since ν0 κ). Furthermore, it is cumbersome to directly compare the
measured probability to the full function (with all the prefactors). It is often easier instead
to make measurements on both the first red sideband and the first blue sideband and then
take the ratio of the two. The constant prefactors disappear in this calculation, and both
functions then have the same experimental signature:
(1)
PGH (∆)
P (1) (∆)
= (1)
= e−2π∆/κ ,
(1)
P (−∆)
PGH (−∆)
(8.40)
which is that of a thermal distribution with temperature T = ~κ/2πkB , which is of the
Gibbons-Hawking form [123] with the expansion rate given by κ. Therefore, although the
Hamiltonian used in the calculations in Ref. [122] was missing terms, the intuition (at
least for a single ion) was correct in that the actual experimental signature in this case
matches that of an ion undergoing thermal motion in a static trap, where the temperature
is proportional to κ.
To see whether this experiment is feasible, we must examine the validity of our approximations. For a typical trap, we expect that ν0 ' 1 MHz, and thus if we take κ ' 1 kHZ, we
easily satisfy the first of conditions (8.25), namely ν0 κ. The second of these conditions
gives a constraint on the modulation time T . For these parameters we expect that T '
a few msec. This is compatible with typical cooling and readout time scales and is less
than those for heating due to fluctuating patch potentials [101]. Thus, this is a realizable
experiment with current technology.
8.4
Conclusion
We have shown that a single trapped ion in a modulated trapping potential can serve as
an experimentally accessible implementation of a quantum harmonic oscillator with timedependent frequency, including robust control over state preparation, manipulation, and
measurement. The ion itself serves both as the oscillating particle and as the local detector
of vibrational motion via coupling to internal electronic states by an external laser. For
the case of a general time-dependent trap frequency, we calculated the first-order excitation
probability for the ion in terms of the solution to the classical equations of motion for the
96
equivalent classical oscillator. We applied this general result to the case of exponential
chirping and corrected the calculation in Ref. [122] for a single ion. We found that while
the results from the two calculations differ, the experimental signature in both cases is the
same and equivalent to that of a thermal ion in a static trap.
97
Chapter 9
Spatial Correlation Functions for
the Collective Degrees of Freedom
of Many Trapped Ions
This chapter presents Ref. [6] in a self-contained fashion.
9.1
Introduction
In contexts as diverse as cosmology, quantum optics and condensed matter physics, spatial correlation functions provide experimental access to important features of spatially
distributed quantum systems. In a cosmological setting, spatial correlations in the temperature distribution of the cosmic microwave background provide direct access to the fluctuations of a primordial quantum field during inflation. Details of these spatial correlations
provide direct and detailed tests of inflationary cosmological models [127]. For example,
Grishchuk [128] has suggested a model in which anisotropies reflect the underlying statistics
of squeezed vacuum states. In condensed matter physics, the change in the length scale of
correlations of the XY model in two dimensions can reveal a Kosterlitz-Thouless transition
and vortex phases [129, 130]. The spatial correlation functions of a Bose-Einstein condensate, as reveled by light scattering from a freely expanding condensate reveal details of the
quantum state of the condensate before expansion [131]. As one example of this, mean field
energy of a condensate is a direct measure of the second order correlation function [132].
The atom loss-rate due to three-body recombination in a BEC is directly related to the
probability of finding three atoms close to each other and can therefore act as a probe of a
third-order correlation function [133]. In this paper we show that similar two-point spatial
correlation functions can be used to probe the collective vibrational state for a string of
trapped ions [101]. Our work is related to that of Franke-Arnold [134], which considers the
spatial coherence properties of just two harmonically trapped particles.
98
Using external lasers, it is possible to couple an internal electronic transition to the
vibrational motion of the ion [121]. Indeed this is how laser cooling of multiple ions to the
vibrational ground state of one or more of their collective normal modes of vibration [102].
Simple quantum information processing tasks can be realised by coupling internal states
through collective motional degrees of freedom [110, 111]. In this paper, we show how the
ability to couple internal and vibrational degrees of freedom, plus the ability to efficiently
read out the internal electronic state using a fluorescence shelving scheme [101], provides a
path to experimentally measuring spatial correlation functions for the collective vibrational
degrees of freedom. This provides a discrete analogue of spatial correlation functions for a
scalar quantum field.
The scheme has two components: (1) external lasers weakly couple the motion of two
distinct ions to their local displacement from equilibrium; subsequently, (2) external readout
lasers are used to probe the electronic state of all the ions. In the second stage, some ions
will “light up” (made more precise below), while others remain dark. Thus, for N ions,
we get an N -element stochastic binary string (e1 , e2 , . . . , eN ), with em = 1 if an ion lights
up and em = 0 otherwise. This is the measurement record, with the ordering of the string
corresponding exactly with the position ordering of the ions in the trap. We can then define
the empirical two-point correlation function Pmn as the classical average Pmn = E[em en ].
Our objective in this paper is to relate this classical correlation function to the quantum
mechanical correlations of the ion displacement amplitudes (e.g., terms such as hq̂m q̂n i)
and, furthermore, to determine characteristic experimental signatures of different collective
quantum states of motion.
9.2
Spatial Correlation Functions
We summarise the standard results for the spatial correlation functions of a scalar field (see,
for example, Ref. [135]). A scalar field φ(x, t) may be expanded as
φ̂(x, t) =
X
ak uk (x, t) + a†k u∗k (x, t) ,
(9.1)
k
where uk (x, t) is a positive-frequency mode function, and u∗k (x, t) is its negative-frequency
counterpart. In the commonly used case of plane waves (with box normalization), we can
define the positive- and negative-frequency components of φ(x, t) as
and
1 X
ak e+i(k·x−ωk t)
ψ(x, t) := √
V k
1 X † −i(k·x−ωk t)
ψ † (x, t) := √
a e
,
V k k
99
(9.2a)
(9.2b)
respectively. In the continuum limit, these operators satisfy the standard boson equal-time
commutation relations:
[ψ(x, t), ψ † (x0 , t)] = δ(x − x0 ) .
(9.3)
With all of the time-dependence in the exponential e±iωk t , we can change to the Schrödinger
picture simply by simply removing this piece (which is equivalent to evaluation at t = 0).
Now in the Schrödinger picture, we define the first- and second-order normally ordered
spatial correlation functions,
D
E
G(1) (x, x0 ) := ψ̂ † (x)ψ̂(x0 )
D
E
G(2) (x, x0 ) := ψ̂ † (x)ψ̂ † (x0 )ψ̂(x0 )ψ̂(x) .
and
(9.4a)
(9.4b)
We also define the normalized correlation functions,
G(1) (x, x0 )
p
g (1) (x, x0 ) := p
G(1) (x, x) G(1) (x0 , x0 )
g (2) (x, x0 ) :=
and
G(2) (x, x0 )
.
G(1) (x, x)G(1) (x0 , x0 )
(9.5a)
(9.5b)
If each mode is independently excited to a thermal state, with no mode-mode correlations,
we can show that [135]
2
g (2) (x, x0 ) = 1 + g (1) (x, x0 ) ,
(9.6)
where the second term represents bosonic bunching. In the continuum limit,
g
(1)
1
(x, x ) =
N
0
Z
0
d3 k n(k)eik·(x−x ) ,
(9.7)
where n(k) represents the thermal occupation number of mode k and N is the total occupation number, given by
Z
N :=
d3 k n(k) .
(9.8)
It is apparent that g (2) (x, x0 ), as a function of (x − x0 ), starts at a value of 2 and decreases
to 1. How fast it decreases is a measure of the range of spatial correlation functions and
depends on the k-dependance of n(k).
The example of the thermal states is a special case of a general result for all classical
states of the field for which
g (2) (x, x) ≥ g (2) (x, x + y)
(9.9)
Nonclassical states violate this inequality. The particular example of squeezed light has been
studied in some detail [136]. The effect of the squeezing is to induce spatial correlations
that modulate an effective thermal background density. In some cases this reduces the
100
correlation function G(2) (x, x0 ) below the thermal value of 2, a phenomenon related to
photon antibunching in the optical case [137]. It can also increase it above 2. The ability to
change the first and second order correlation functions in this way is what lies behind the
new field of quantum imaging [138]. In Section 9.6, we will contrast thermal and squeezed
states in the case of vibrations in a linear ion trap and show that the latter have a similar
nonclassical signature.
9.3
9.3.1
Measurement of Ion Trap Spatial Correlations
Normal modes of vibration
Linearizing the potential created by the overall harmonic potential due to the trap electrodes
and the mutual Coulomb repulsion between the ions, we can model the collective motion
of the ions in a linear trap as a collection of coupled harmonic oscillators. With a coupling
matrix A obtained from the linearized combination of these potentials, the Hamiltonian is
given by [102]
H0 =
1 T
M ν2 T
p p+
q Aq ,
2M
2
(9.10)
where q = (q1 , . . . , qN )T is a column vector of operators corresponding to the displacement of
each ion from its equilibrium position, p = (p1 , . . . , pN )T is a column vector of corresponding
momenta, M is the mass of each ion, and ν is the effective harmonic trap frequency provided
by the trap electrodes (typically, ν ∼ a few MHz [101]). The coupling matrix A is symmetric
and positive-definite. Thus, we can diagonalize it:
A = BT ΛB ,
(9.11)
where Λ = diag(µ1 , . . . , µN ) is a diagonal matrix of positive eigenvalues, arranged in ascending order. The orthogonal matrix B defines the transformation to normal-mode coordinates Q = Bq and momenta P = Bp. Equation (9.10) can be rewritten in these coordinates
as
1 T
M ν2 T
P P+
Q ΛQ
2M
2
N
X
Pp2
M νp2 2
=
+
Qp
2M
2
p=1
X
=
~νp a†p ap ,
H0 =
p
101
(9.12)
where
√
νp := ν µp
(9.13)
is the oscillation frequency of normal mode p, and (since the normal modes oscillate independently) we have diagonalized this Hamiltonian using the standard prescription for a
set of independent harmonic oscillators, ignoring the zero-point energy. The normal-mode
raising and lowering operators satisfy
[ap , a†p0 ] = δpp0 ,
(9.14)
as is appropriate for independent bosonic modes.
The local oscillations qm about the ions’ equilibrium positions are given, in the interaction picture, by [102]
qm (t) =
N
X
p=1
s
~
b(p) (ap e−iνp t + a†p eiνp t ) ,
2M νp m
(9.15)
(p)
where m ∈ {1, . . . , N } labels the ion, and bm is an entry of B, which defines the spatial mode
functions of the normal mode.1 We can define a unitless version of this local displacement
oeprator, as well:
φm (t) :=
N
(p)
X
bm
(e−iνp t ap + eiνp t a†p ) .
1/4
µ
p=1 p
(9.16)
The connection between the two is given by
r
qm (t) =
~
φm (t) .
2M ν
(9.17)
The positive- and negative-frequency components of φm (t) may be defined analogously to
those of Eqs. (9.2):
N
(p)
X
bm −iνp t
ψm (t) :=
e
ap
µ 1/4
p=1 p
and
†
ψm
(t)
N
(p)
X
bm +iνp t †
:=
e
ap .
µ 1/4
p=1 p
(9.18)
Because the coupling matrix A only approximates a simple “balls on a string” coupling
(characteristic of a bosonic field), the equal-time commutation relations for these operators
1
Prefactor and phase conventions for qm vary in the literature.
102
only approximate those of a boson field (compare with Eq. (9.3)):
[ψm (t), ψn† (t)]
N
(p) (p)
X
bm bn
=
= A−1/2
.
√
µp
mn
(9.19)
p=1
Still, the fact that the ions’ normal-mode operators commute properly, as in Eq. (9.14),
means that we don’t need to worry about this, as long as we do our calculations using the
normal modes explicitly.
9.3.2
Laser-induced coupling of vibrational and electronic states
The abstract relations between the field spatial correlation functions described in Section 9.2
ultimately are manifest in the observed correlations in spatially distributed detectors of
some kind. In the case of quantum optics, for example, one imagines that the field falls on
a photodetector array. Each element of the array produces a photocurrent I(x, t) indexed
by the position of that photodetector in the array. We can then look at cross-correlations
between photo currents from different detectors in the array. It is the the objective of
measurement theory to determine how those spatially dependent current-current correlation
functions are determined at a fundamental level by the field spatial correlation functions
themselves. Explicit formulae are given in Kolobov [136] using the standard quantum optics
theory of photodetection.
Our measurement model is different from that considered in quantum optics and so
we now explicitly make the connection between the observations and the underlying field
correlation functions for the motion of the ions in the trap. The feature of our model is
that the internal electronic state of each ion can be turned into a local detector for the
displacement of that ion. To achieve this for a given ion, its internal state must become
correlated with its linear displacement from equilibrium.
This correlation is provided by an external laser, which is used to drive an electronic transition between two meta-stable electronic levels |gi and |ei, separated in energy by ~ωA [101,
102, 124]. The interaction between an external classical laser field and the mth ion is described, in the dipole and rotating-wave approximation, by the interaction-picture Hamiltonian [101, 102]
(m)
HI
h
i
(m)
(m)
= −i~Ω0 σ+ (t)eik cos θqm (t) − σ− (t)e−ik cos θqm (t) ,
(9.20)
where Ω0 is the Rabi frequency for the laser-atom interaction (typically, Ω0 ∼ 100 kHz [101]),
k is the magnitude of the laser’s wave vector k, which makes an angle θ with the trap
(m)
axis, qm (t) is the interaction-picture position operator for the mth ion, and σ± (t) are its
interaction-picture electronic raising and lowering operators. Explicitly,
(m)
(m)
σ± (t) = e±i∆t σ±
103
,
(9.21)
(m)
where σ+
(m)
= |eimmhg| and σ−
= |gimmhe|, and
∆ = ωA − ωL
(9.22)
is the detuning of the laser below the atomic transition. The size of the rms fluctuation
in qm as compared to the wavelength of the laser is measured by the Lamb-Dicke parameter
r
η :=
where
(m)
~k 2 cos2 θ
∆xrms
∼
(2π cos θ) ,
2M ν
λL
(9.23)
p
~/2M ν is the rms fluctuation of the center-of-mass mode of the ions in the ground
state. This quantity is representative of the overall rms fluctuations of the mth ion, de(m)
noted ∆xrms , since the frequencies νp for all higher normal modes of the ions remain within
an order of magnitude of ν for small numbers of ions (up to N ∼ 10), realistic for current
experiments [102]. Typical values of the Lamb-Dicke parameter are η ∼ 0.01 to 0.1 [54].
When η 1, the so-called “Lamb-Dicke limit,” the ion is well localized with respect to
the wavelength of the laser, and we can expand the exponentials in Eq. (9.20) to first order
in η, which is equivalent to first order in k cos θ qm (t), giving
(m)
HI
(t) ' ~Ω0 σy(m) (t) + ~Ω0 k cos θ qm (t)σx(m) (t) ,
{z
}
| {z } |
sideband
carrier
(m)
(m)
where σx (t) = e+i∆t σ+
(9.24)
(m)
(m)
(m)
+ e−i∆t σ− , and σy (t) = −ie+i∆t σ+
(m)
+ ie−i∆t σ− . The
first term corresponds to excitation of the transition directly by the laser, while the second
couples the atomic transition to vibrational motion.
The “carrier” term is resonant when ∆ = 0 and corresponds to direct excitation of the
atomic transition, which does not couple to the vibrational motion at all. For sufficiently
long-time detection, another rotating-wave approximation may be made in which only resonant (nonoscillatory) terms are kept. In this case, if the carrier transition is sufficiently
off-resonant (∆ 6= 0), then it can be neglected, leaving
(m)
HI
sideband
N
(r)
X
bm
(∆ 6= 0)
1/4
r=1 µr
(t) −−−−−→ ~Ω0 η
(m)
(m)
(e−iνr t ar + eiνr t a†r )(ei∆t σ+ + e−i∆t σ− ) .
(9.25)
This remaining “sideband” term can be used to couple the atomic transition to the vibrational motion through a judicious choice of detuning, ∆ = ±νp , for some normal mode
frequency νp . The first red sideband transition is obtained by setting ∆ = νp ; that is, the
laser is detuned one unit of vibrational energy below (to the red of) the atomic transition.
104
In this case, the resonant terms are
(m)
HI
(p)
red sideband, mode p
bm
(∆ = νp )
1/4
µp
(t) −−−−−−−−−−−−−→ ~Ω0 η
(m)
(m)
(ap σ+ + a†p σ− ) .
(9.26)
This is of Jaynes-Cummings form [101, 103], corresponding to excitation of the atomic transition with energy gap ~ωA = ~ωL + ~νp upon absorption of one laser photon at energy ~ωL ,
along with absorption of one vibrational phonon at energy ~νp . Detuning the laser above
(to the blue of) the atomic transition by one unit of vibrational energy (∆ = −νp , for some
mode p) generates the blue sideband transition, which has resonant terms
blue sideband, mode p
(m)
HI (t) −
−−−−−−−−−−−−−
→
(∆ = −νp )
(p)
~Ω0 η
bm
1/4
µp
(m)
(m)
(a†p σ+ + ap σ− ) .
(9.27)
This corresponding to atomic excitation of the transition at energy ~ωA = ~ωL − ~νp upon
absorption of one laser photon at energy ~ωL , along with emission of one vibrational phonon
at energy ~νp .
The first red-sideband transition uses the ion itself as its own detector of motional quanta
and this configuration comprises the first half of our two-stage detection model. As long as
the coupling strength Ω0 η is weak enough, no more than a single |gi → |ei transition will
be excited for a given ion in the time during which the laser coupling is active. Although
the approximations above simplify the interaction Hamiltonians and provide a means to
understand the physical mechanisms at work, for finite detection times the full form of the
interaction may be needed. As such, we will retain the “sideband” term of Eq. (9.24) as
the interaction Hamiltonian, which has the form of a De Witt monopole coupling [105], in
order to allow for greater applicability of our results. In the examples given in Section 9.6,
we will apply the long detection-time approximation at the end of the calculations.
9.3.3
Excitation probabilities and correlation functions
In the interaction picture, the full interaction Hamiltonian is
HI (t) := ~Ω0 η
X0
φm (t)σx(m) (t) ,
(9.28)
m
where the prime on the sum indicates that only the ions being addressed by interaction
lasers are included, and the unitless displacement operator φm (t) is defined in Eq. (9.16).
The time evolution operator may be written formally as
U (T ) := T
Z
−i T
exp
dt HI (t)
,
~ 0
105
(9.29)
where T is the Dyson time-ordering symbol. The time-ordered exponential is defined by
the time-ordering of its Taylor-series expansion, the first few terms of which are written
here:
U (T ) = 1 +
−i
~
Z
T
dt HI (t) +
0
1
2!
−i
~
2 Z
T
Z
T
dt2 T {HI (t1 )HI (t2 )} + · · · .
dt1
0
(9.30)
0
We will use these three terms in what follows.
As discussed in the introduction, we wish to calculate correlation functions for simultaneous detected excitations. Detecting the excitation of a given ion m corresponds to
measuring the projector
Pm := |eimmhe| ,
(9.31)
where |eim is the excited electronic state of the mth ion. Pm acts trivially on all other
electronic states and on all vibrational modes. In the experiment, the electronic state of
the ion is determined using the technique of fluorescence on a cycling transition [101]. The
excited state |eim is caused to make a dipole allowed transition to another auxiliary level
which then decays back to the state |eim through spontaneous emission. If this transition
is saturated, a very large fluorescent photon flux is easily detected. The measurement very
nearly approaches a projection measurement of the operator Pm with an efficiency greater
than 99%.
These projectors commute for all m, so we can represent joint detection of the excitation
of multiple ions as
Pm1 ···mM := Pm1 · · · PmM .
(9.32)
We will be interested here in simultaneous detection on at most two ions,
Pmn = |eimmhe| ⊗ |einnhe|
(9.33)
(assuming m 6= n, since Pmm is just Pm ), and initial states (in the interaction picture, at
time t = 0) of the form
ρ0 = ρ ⊗ |gihg|⊗N ,
(9.34)
where ρ is the vibrational state, and all of the ions are in the ground electronic state.
The ions are assumed to be ordered in a linear array. Thus the subscripts mn are an
implicit spatial index. This is indicated schematically in Figure 9.1. The probability for
measuring ion m in an excited electronic state after the interaction Hamiltonian is applied
106
qm
(a) Coupling stage
xN
xm
x1
x
(b) Readout stage
0
0
0
1
xN
xm
x1
0
x
Figure 9.1: An illustration of the linear ion array with N ions. The index m runs from 1
to N . This may be converted to a position label xm as indicated, which labels the equilibrium
position of each ion. The quantum degree of freedom, q̂m describes small displacements from
equilibrium. In (a) weak lasers couple the internal electronic state of the ion to the displacement
from equilibrium of that ion. In (b) strong readout lasers drive fluorescence conditional on the
electronic state of the ion. If an a given ion is in the excited state, it fluoresces (giving the
result 1) or it does not fluoresce (giving the result 0).
from time t = 0 to time t = T is given by
Pm := hU † (T )Pm U (T )i ,
(9.35)
where the expectation value is taken with respect to ρ0 . Similarly, the probability that both
ions m and n are in the excited electronic state after the evolution is
Pmn := hU † (T )Pmn U (T )i .
(9.36)
Since the only outcomes of an actual measurement is a stochastic binary string indicating
which ions lit up and which did not (e1 , . . . , eN ), these probabilities correspond directly to
classical expectation values over the entries em in this string:
E[em ] = Pm ,
E[em en ] = Pmn ,
(9.37)
Of course, other correlation functions (for three or more ions) may be defined analogously.
9.4
Excitation Probability Calculations for General States
We proceed to calculate Pm and Pmn to second order in the Dyson series expansion,
Eq. (9.30). The coupling lasers are assumed to be weak enough to justify this pertur(0)
bative treatment. The zeroth-order approximation Pm vanishes since the initial state has
107
no ions excited. The first-order term is
(1)
Pm
1
= 2
~
Z
T
0
= (Ω0 η)2
Z
T
dt2 hHI (t1 )Pm HI (t2 )itotal
Z T
dt1
dt2 mhg|σx(m) (t1 )|eimmhe|σx(m) (t2 )|gim hφm (t1 )φm (t2 )i ,
dt1
Z T
0
0
(9.38)
0
resulting in
(1)
Pm
2
T
Z
= (Ω0 η)
Z
T
dt1
0
dt2 e−i∆(t1 −t2 ) hφm (t1 )φm (t2 )i ,
(9.39)
0
where expectation values are now over the vibrational modes only unless otherwise indicated.
(2)
(m) (m)
The second-order term Pm = 0 because mhg|HI (t1 )HI (t2 )|eim ∼ mhg|σ± σ± |eim = 0—
i.e., no two applications of the electronic raising/lowering operators can lower the excited
state to the ground state.
We’re now prepared to tackle the spatial correlation function (9.36). Once again, the
(0)
zeroth-order term Pmn = 0 because no ions are electronically excited to begin with. Also,
considering that Pmn projects onto the excited electronic states of two distinct ions while
(1)
each HI (t) can only raise one ion at a time, mhg|⊗ nhg|HI (t1 )|eim ⊗|ein = 0, making Pmn = 0,
as well. Therefore, we must go to second order:
(2)
Pmn
Z T
Z T
Z T
Z T
1
= 4
dt1
dt2
dt3
dt4 hT̄ {HI (t2 )HI (t1 )}Pmn T HI (t3 )HI (t4 ) itotal
4~ 0
0
0
0
Z T
1
(m)
(n)
(m)
(n)
d4 t hT̄ [HI (t2 ) + HI (t2 )][HI (t1 ) + HI (t1 )] Pmn
= 4
4~ 0
(m)
(n)
(m)
(n)
× T [HI (t3 ) + HI (t3 )][HI (t4 ) + HI (t4 )] itotal
Z T
1
(m)
(n)
(n)
(m)
= 4
d4 t hT̄ {HI (t2 )HI (t1 ) + HI (t2 )HI (t1 )}Pmn
4~ 0
(m)
× T {HI
(n)
(n)
(m)
(t3 )HI (t4 ) + HI (t3 )HI
(t4 )}itotal .
(9.40)
The antitime-ordering symbol T̄ acts like T but instead orders the terms from earliest to
latest. Trading integration variables (t1 ⇔ t2 and/or t3 ⇔ t4 ) and minding the (anti)timeordering, we can simplify this to
(2)
Pmn
1
= 4
~
Z
0
T
(m)
d4 thT̄ {HI
(n)
(m)
(t2 )HI (t1 )}Pmn T {HI
108
(n)
(t3 )HI (t4 )}itotal .
(9.41)
The terms from the Hamiltonians that survive the projection and contraction with the
electronic ground state are
(m)
(t2 )HI (t1 )} =⇒ (~Ω0 η)2 T̄ {φm (t2 )φn (t1 )}e−i∆t2 e−i∆t1 σ− σ−
(m)
(t3 )HI (t4 )} =⇒ (~Ω0 η)2 T {φm (t3 )φn (t4 )}e+i∆t3 e+i∆t4 σ+ σ+ . (9.43)
T̄ {HI
T {HI
and
(n)
(m) (n)
(n)
(m) (n)
(9.42)
Plugging these in gives
(2)
Pmn
4
Z
T
= (Ω0 η)
d4 t e−i∆(t1 +t2 −t3 −t4 ) hT̄ {φm (t2 )φn (t1 )}T {φm (t3 )φn (t4 )}i .
(9.44)
0
Several ways exist for expressing the four-point correlation function in Eq. (9.44) in a
convenient form by use of Wick’s theorem. These are included in the Appendix. One form
is particularly useful, though, and that it applies when the state has a Wigner function
that is a Gaussian with zero mean (i.e., a “Gaussian state”). In this case, we may write
(repeated from Eq. (9.115))
Gaussian
hT̄ {φm (t2 )φn (t1 )}T {φm (t3 )φn (t4 )}i −−−−−→ hφm (t2 )φm (t3 )ihφn (t1 )φn (t4 )i
state
+ hφm (t2 )φn (t4 )ihφn (t1 )φm (t3 )i + hT̄ {φm (t2 )φn (t1 )}ihT {φm (t3 )φn (t4 )}i . (9.45)
Comparing this with Eq. (9.39), we see that the first term will always give simply Pm Pn ,
and the second will always give a term similar to this, but with a different geometric factor.
Thus, once we have Pm , we need only ever explicitly calculate the last two terms from
Eq. (9.45) to get Pmn .
9.4.1
Long-time interaction
Before moving on, let’s have a look at what affect a long-time interaction would have on the
evolution. In this case, we can employ the rotating wave approximation and use Eq. (9.26)
as the interaction Hamiltonian. In this case, Eq. (9.29) can be evaluated to
#
X0 b(p)
m
(m)
(m)
(a σ
+ a†p σ− ) .
U (T ) −−−−−→ exp −iT (Ω0 η)
1/4 p +
(∆ = νp )
m µp
"
RWA
(9.46)
There is a balance to be maintained, here, though. On the one hand, the detection time T
needs to be long enough so that the rotating wave approximation is valid. This is the
requirement that T ν −1 (and we assume that ∆ ∼ ν). On the other hand, it needs to be
short enough so that we can use perturbation theory, which requires T (Ω0 η)−1 . When
both of these conditions are satisfied simultaneously, that is,
ν −1 T (Ω0 η)−1 ,
109
(9.47)
we can expand the exponential in Eq. (9.46), as in Eq. (9.30). This gives
RWA
(1)
Pm
−−−−−→
(∆ = νp )
(p)2 D
E
T2
2
2 bm
†
hH
P
H
i
=
T
(Ω
η)
a
a
,
√
0
p
I m I total
~2
µp p
(9.48)
which agrees with what is obtained from selecting only the resonant terms directly from
Eq. (9.39). A similar result holds for the second-order function:
RWA
(2)
Pmn
−−−−−→
(∆ = νp )
(p) (p) 2 D
E
T4 2
4
4 (bm bn )
2
† †
=
T
(Ω
η)
H
P
H
a
a
a
a
.
p
p
0
mn
I
I
p
p
total
4~4
µp
(9.49)
If we were to allow for different detunings on each ion, ∆m = νp , ∆n = νp0 , then this result
could be generalized:
(2)
Pmn
(p) (p0 ) 2 D
E
T4 2
2
4
4 (bm bn )
a†p a†p0 ap0 ap . (9.50)
−−−−−−−−−−−−−→ 4 HI Pmn HI total = T (Ω0 η) √
µp µp0
(∆m = νp , ∆n = νp0 ) 4~
RWA
This interaction would require that a second laser beam be detuned from the first, an
experimental complication we won’t consider further.
For typical values of the parameters (ν ∼ 1 MHz, Ω0 ∼ 100 kHz, and ν ∼ 0.01),
Condition (9.47) requires that 10−6 s T 10−3 s, a rather narrow band in which
to operate well within both regimes. Given this limitation, in the next section, when we
calculate Pm and Pmn for any Gaussian state as a function of its covariance matrix, we
will retain the perturbative condition T (Ω0 η)−1 , but we will not use the rotating wave
approximation, in order to allow for a more general class of interactions, including ones for
which T . ν −1 . We will also assume that both interaction lasers have the same detuning ∆
for experimental simplicity.
9.5
9.5.1
Evaluation for Gaussian States
Two-point functions
As discussed in the Appendix, when the vibrational state is a zero-mean Gaussian, all
measured correlation functions can be evaluated from the two-point functions hφm (t)φn (t0 )i
and hT {φm (t)φn (t0 )}i. Let’s define the two-time-dependent matrices
and
Υ(t, t0 ) := φ(t)φ(t0 )T
Υ̊(t, t0 ) := T {φ(t)φ(t0 )T }
(9.51)
(9.52)
(9.53)
110
for which Υmn (t, t0 ) = hφm (t)φn (t0 )i and Υ̊mn (t, t0 ) = hT {φm (t)φn (t0 )}i. It’s easy to see
from the entries that

Υ(t, t0 )
0
Υ̊(t, t ) =
Υ(t, t0 )∗
if t > t0 ,
if t < t0 ,
(9.54)
since φm (t) is Hermitian. Thus, we really only need to worry about Υ(t, t0 ) for the moment.
Using Eqs. (9.16) and (9.11), we can transform to the normal-mode two point function
instead:
Υ(t, t0 ) = BT Λ−1/4
D
T E −1/4
Λ
B,
a(t) + ã(t) a(t0 ) + ã(t0 )
(9.55)
where a(t) := (a1 (t), . . . , aN (t))T is a column vector of interaction-picture lowering operators for the normal modes, ã(t) := (a†1 (t), . . . , a†N (t))T is the equivalent column vector of
interaction-picture raising operators, and B and Λ are defined through Eq. (9.11). We can
write the time-dependence of these vectors in a very compact form:
a(t) = E(t) ◦ a ,
(9.56)
E(t) := (e−iν1 t , . . . , e−iνN t )T
(9.57)
where
is a column vector of time-dependent coefficients, and the symbol ◦ represents the Hadamard
product (element-wise multiplication). Similarly,
ã(t) = E(−t) ◦ ã .
(9.58)
Any Gaussian state with zero mean is uniquely defined by its covariance matrix. While
many varieties of covariance matrix can be defined [54], an obvious choice here would be
to use the two matrices aaT and ãaT . We can get the other combinations by noting
∗
that ããT = aaT , and aãT = ãaT + 1. Thus, we can go one step further and define
a matrix of coefficients
E(t, t0 ) := E(t)E(t0 )T ,
(9.59)
0
such that Ers (t, t0 ) = e−i(νr t+νs t ) . Using this shorthand, we can isolate the time dependence
from the expectation value in Eq. (9.55) and write the result in terms of the initial covariance
111
matrix:
D
T E
a(t) + ã(t) a(t0 ) + ã(t0 )
= aaT ◦ E(t, t0 ) + aãT ◦ E(t, −t0 ) + ãaT ◦ E(−t, t0 ) + ããT ◦ E(−t, −t0 )
= 2 Re aaT ◦ E(t, t0 ) + ãaT + 1 ◦ E(t, −t0 ) + ãaT ◦ E(−t, t0 ) .
(9.60)
K(t, t0 ) :=
Considering Eq. (9.54), we’re going to need the complex conjugate of Eq. (9.60). Elementby-element evaluation will reveal the following equivalences:
T ∗ aã ◦ E(t, −t0 ) = ãaT + 1 ◦ E(−t, t0 ) ,
T ∗ ãa ◦ E(−t, t0 ) = aãT − 1 ◦ E(t, −t0 ) ,
(9.62)
∗
T aã ◦ E(t, −t0 ) + ãaT ◦ E(−t, t0 )
= aãT ◦ E(t, −t0 ) + ãaT ◦ E(−t, t0 ) + 1 ◦ E(−t, t0 ) − E(t, −t0 )
= ãaT + 1 ◦ E(−t, t0 ) + ãaT ◦ E(t, −t0 ) .
(9.63)
(9.61)
leading to
This is the same as the last two terms in Eq. (9.60), except that the E-matrices have been
exchanged. Therefore, we can define

E(t, −t0 )
E̊(t, t0 ) :=
E(−t, t0 )
if t > t0 ,
if t < t0 .
(9.64)
Notice the subtle difference between Υ̊(t, t0 ) and E̊(t, t0 ). The ring-notation is consistent
with its purpose—to describe a function associated with time-ordering—but not necessarily
with the details of the definition. Recalling Eq. (9.54), the time-ordered expectation value
can now be written succinctly:
D n
T oE
K̊(t, t0 ) := T
a(t) + ã(t) a(t0 ) + ã(t0 )
= 2 Re aaT ◦ E(t, t0 ) + ãaT + 1 ◦ E̊(t, t0 ) + ãaT ◦ E̊(t, t0 )∗ .
(9.65)
Pluggin into Eqs. (9.55) and (9.54) gives
Υ(t, t0 ) = BT Λ−1/4 K(t, t0 )Λ−1/4 B ,
0
T
Υ̊(t, t ) = B Λ
−1/4
0
K̊(t, t )Λ
−1/4
with K(t, t0 ) defined in Eq. (9.60) and K̊(t, t0 ) in Eq. (9.65).
112
B,
(9.66)
(9.67)
9.5.2
Probabilities in terms of the covariance matrix
We have successfully consolidated all of the time-dependence into E(t, t0 ) and E̊(t, t0 ). This
makes evaluation of Eqs. (9.39) and (9.44) dependent only on integrals involving elements
of these matrices. The following integral will be useful:
Z
1
T
T /2
ωT
= sinc
2
iωt
,
(9.68)

x−1 sin x if x 6= 0 ,
sinc x :=
1
if x = 0 ,
(9.69)
dt e
−T /2
where
and we have symmetrized the limits of integration by setting the laboratory clock appropriately—
a passive operation that does not affect the physics of the experiment. Eq. (9.68) is a
bandwidth-limited Fourier transform. In anticipation of future calculations, let’s define
Z
1
Srs (ω, ω ) := 2
T
0
T /2
Z
T /2
0 0
dt0 ei(ωt+ω t ) Ers (t, t0 )
−T /2
−T /2
0
(ω − νs )T
(ω − νr )T
sinc
.
= sinc
2
2
dt
(9.70)
We can now collect the Srs (ω, ω 0 ) elements into a matrix S(ω, ω 0 ). We can also define
1
S̊rs (ω, ω ) := 2
T
0
Z
T /2
Z
T /2
dt
−T /2
0 0
dt0 ei(ωt+ω t ) E̊rs (t, t0 ) ,
(9.71)
−T /2
and a corresponding matrix S̊(ω, ω 0 ), although we will leave it unevaluated for now.
(1)
Using these tools, let’s calculate Pm :
(1)
Pm
2
Z
T /2
= (Ω0 η)
Z
−T /2
=
T /2
dt2 e−i∆(t1 −t2 ) Υmm (t1 , t2 )
dt1
−T /2
(Ω0 η)2 eTm BT Λ−1/4
= T 2 (Ω0 η)2 eTm BT Λ
"Z
T /2
Z
dt1
−T /2
−1/4
#
T /2
−i∆(t1 −t2 )
dt2 e
K(t1 , t2 ) Λ−1/4 Bem
−T /2
◦ S(−∆, ∆) + ããT ◦ S(∆, −∆)
T T
+ ãa + 1 ◦ S(−∆, −∆) + ãa ◦ S(∆, ∆) Λ−1/4 Bem ,
aa
T
(9.72)
where em is a unit column vector used (twice) to pick out the correct element from the
matrix. We should also point out that Eq. (9.72) also holds for non-Gaussian states, for
which a covariance matrix can also be defined even though it does not specify the state
completely.
113
The correlation function Pmn can be calculated similarly. In this case, it is important
that the state be (zero-mean) Gaussian because we will use the simplification provided by
Eq. (9.45). As already discussed,
(Term 1) = Pm Pn .
(9.73)
The second is almost the same, except it involves a different element of Υ(t, t0 ). An analogous
calculation to the one above shows that
(Term 2) = T
4
(Ω0 η)4 eTm BT Λ−1/4
aaT ◦ S(−∆, ∆) + ããT ◦ S(∆, −∆)
2
T T
−1/4
+ ãa + 1 ◦ S(−∆, −∆) + ãa ◦ S(∆, ∆) Λ
Ben , (9.74)
the only difference (besides the squaring) being the presence of en at the end, instead
of em . The third term is more complicated, due to the time ordering. Nevertheless, it can
be written
Z
2
Z T /2
T /2
(Term 3) = (Ω0 η)4 dt1
dt2 ei∆(t1 +t2 ) Υ̊mn (t1 , t2 )
−T /2
−T /2
2
"Z
#
Z T /2
T /2
4 T T −1/4
i∆(t1 +t2 )
−1/4
= (Ω0 η) em B Λ
dt1
dt2 e
K̊(t1 , t2 ) Λ
Ben −T /2
−T /2
= T 4 (Ω0 η)4 eTm BT Λ−1/4 aaT ◦ S(∆, ∆) + ããT ◦ S(−∆, −∆)
2
T T
∗
−1/4
+ ãa + 1 ◦ S̊(∆, ∆) + ãa ◦ S̊(−∆, −∆) Λ
Ben . (9.75)
Evaluating this term boils down to evaluating Eq. (9.71). The sum of Terms 1, 2, and 3
(2)
gives Pmn for a zero-mean Gaussian state.
9.6
Examples
In order to demonstrate the usefulness of the correlation function, we’ll compare a thermal
state with a given average phonon number to a corresponding uniformly squeezed state with
the same average phonon number. We will also assume that the interaction is active on a
timescale long compared to the period of vibrations of the ions—that is, T ν −1 . In both
cases, the probability of excitation for any given ion is the same, but, as we shall see, the
correlations are stronger in the squeezed state versus the corresponding thermal state. Our
measure of correlations will be
fmn :=
E[em en ]
Pmn
=
,
Pm Pn
E[em ]E[en ]
114
(9.76)
which is a normalized correlation function akin to those defined in Eqs. (9.5) but is based
on detection probabilities, rather than underlying quantum correlations. The connection to
the quantum state, of course, was made in the preceding sections.
9.6.1
Thermal state
A thermal state at temperature τ is a zero-mean Gaussian state in which
har as i = 0 ,
D
E
a†r as = n̄r δrs ,
(9.77)
(9.78)
where
nr :=
1
eβ~νr
(9.79)
−1
is the average number of phonons in mode r, and β = (kB τ )−1 . Plugging into Eq. (9.72)
gives
(1)
Pm
= T 2 (Ω0 η)2
N
(p)2
X
o
bm n
2
2
T
T
.
(n̄
+
1)
sinc
(∆
+
ν
)
+
n̄
sinc
(∆
−
ν
)
√
p
p 2
p
p 2
µp
(9.80)
p=1
The sinc2 -function is sharply peaked at ∆ = ±νp , and in the long-time limit (T ν −1 ),
we have
(T ν −1 )
(1)
Pm
−−−−−−→ T 2 (Ω0 η)2
n
o
X b(p)2
m
2
2
(n̄
+
1)(2π)
δ
+
n̄
(2π)
δ
√
p
p
(∆+νp )
(∆−νp ) ,
µ
p
p
(9.81)
where the Kronecker-δ symbol satisfies δ0 = 1 and δx = 0 for x 6= 0 to within a bandwidth
of approximately T −1 . This is gives the same results as would be obtained after using
the rotating wave approximation, as described in Section 9.4.1. Choosing a particular
normal mode frequency as the detuning (∆ = νp ) accords with the result calculated directly
−1/2 (p)
bm ,
from Eq. (9.48). In this case, the detection probability has a geometric factor µp
corresponding to the position of the ion within the normal mode p being addressed and is
proportional to the average number of phonons n̄p in the mode for red sideband detuning
(and n̄p + 1 for the blue sideband), as expected.
(2)
Moving on to the correlation probability Pmn , we can evaluate Eq. (9.74) to
(Term 2) =
"
4
4
T (Ω0 η)
#2
(p) n
X b(p)
o
m bn
2
2
T
T
(n̄p + 1) sinc (∆ + νp ) 2 + n̄p sinc (∆ − νp ) 2
. (9.82)
√
µp
a
115
(1)
In the long interaction time-limit, this term behaves similarly to Pm :
(T ν −1 )
"
4
4
(Term 2) −−−−−−→ T (Ω0 η)
#
(p) n
o 2
X b(p)
m bn
2
2
(n̄p + 1)(2π) δ(∆+νp ) + n̄p (2π) δ(∆−νp )
.
√
µp
a
(9.83)
In order to evaluate Eq. (9.75), we need to evaluate S̊(±∆, ±∆) from Eq. (9.71). Using
Eq. (9.78), E̊(t1 , t2 ) is diagonal, meaning we only need to calculate
1
T2
Z
1
T2
Z
2
= 2
T
Z
S̊pp (±∆, ±∆) =
=
T /2
T /2
Z
dt1
−T /2
T /2
−T /2
T /2
Z
dt1
−T /2
dt2 e±i∆(t1 +t2 ) E̊pp (t1 , t2 )
dt2 e±i∆(t1 +t2 ) e−iνp |t1 −t2 |
−T /2
T /2
t
Z
dt2 e±i∆(t1 +t2 ) e−iνp (t1 −t2 ) .
dt1
(9.84)
−T /2
−T /2
Since T ν −1 ∼ ∆, we can change the integration limits with T → ∞:
2
S̊pp (±∆, ±∆) = 2
T
Z
∞
t
Z
dt2 e±i∆(t1 +t2 ) e−iνp (t1 −t2 ) .
dt1
−∞
(9.85)
−∞
With the integration now over the entire half-plane defined by t2 < t1 , we can rotate our
integration axes using
1
u = (t1 + t2 )
2
1
v = (t1 − t2 )
2
t1 = u + v
t2 = u − v
(9.86)
to obtain
∞
Z
S̊pp (±∆, ±∆) = 2
Z
∞
dv
0
du e±i2∆u e−i2νp v ∼ δ∆ δνp → 0 .
(9.87)
−∞
Thus, in the thermal case,
(T ν −1 )
(Term 3) −−−−−−→ 0 .
(9.88)
Consolidating these results, we have
(2)
fmn '
Pmn
(1)
(1)
=2,
(9.89)
P m Pn
for any detuning ∆ = ±νp , any nonzero temperature, and any choice of ions (m, n) to
measure. This is consistent with the expected results for a second-order correlation function
for a thermal state [54].
116
9.6.2
Uniformly squeezed normal modes
The state to be considered in this section is uniformly squeezed in all normal modes. Such
a state has
p
har as i = − n̄(n̄ + 1) δrs ,
D
E
a†r as = n̄ δrs ,
(9.90)
(9.91)
where n̄ = sinh2 r is the mean phonon number of each mode (assumed the same for each
mode) as a function of the squeezing parameter r.
(1)
Proceeding as for the thermal state, we have nearly the same expression for Pm :
(1)
Pm
N
(p)2
X
bm n
(n̄ + 1) sinc2 (∆ + νp ) T2 + n̄ sinc2 (∆ − νp ) T2
= T (Ω0 η)
√
µp
p=1
p
o
. (9.92)
− 2 n̄(n̄ + 1) sinc (∆ − νp ) T2 sinc (∆ + νp ) T2
2
2
In the long detection time-limit, however, the last term makes no difference, and
(T ν −1 )
(1)
(1)
Pm,squeezed −−−−−−→ Pm,thermal ,
(9.93)
(1)
where Pm,thermal is Eq. (9.80) for a thermal state with temperature chosen to make n̄ = n̄p
in Eq. (9.79) for a chosen detuning of ∆ = νp .
(2)
An analogous calculation for the second term of Pmn shows that
"
(Term 2) = T 4 (Ω0 η)4
(p) n
X b(p)
m bn
(n̄ + 1) sinc2 (∆ + νp ) T2 + n̄ sinc2 (∆ − νp ) T2
√
µp
a
#2
p
o
T
T
. (9.94)
− 2 n̄(n̄ + 1) sinc (∆ − νp ) 2 sinc (∆ + νp ) 2
In the limit of T ν −1 , this gives no change from the same term in the equivalent thermal
case:
(T ν −1 )
(Term 2)squeezed −−−−−−→ (Term 2)thermal ,
(9.95)
In the thermal case, the third term vanished in the long detection time-limit. In the squeezed
case, it does not, and this generates the difference in the correlation functions:
(T ν −1 )
"
4
4
(Term 3) −−−−−−→ T (Ω0 η)
#
(a)
n
o 2
X b(a)
m bn p
n̄(n̄ + 1) δ(∆+νa ) + δ(∆−νa )
.
√
µa
a
117
(9.96)
Figure 9.2: Correlation functions Pmn for the normal modes of 10 ions in the case of the
examples considered in Section 9.6. The top row represents Pmn for a detuning corresponding
to normal modes 1 through 5; the bottom row, modes 6 through 10. Since Pmn ∝ Pm Pn for
both types of states, except in the case of the center-of-mass mode (mode 1), which is uniform
anyway, the colors are normalized such that white represents the maximum value, and black
represents the minimum in each case. Thermal and uniformly squeezed states both give the
same results under this normalization. The diagrams show two things: (1) that the structure
of the modes revealed by the correlations, and (2) which pairs of ions (white boxes) are most
useful for measurement when detuning to a given mode frequency.
This shows that even though local measurements have the same excitation statistics as the
equivalent thermal state, simultaneous measurements of two ions do not—the correlations
are stronger in the squeezed case, as should be expected. For the uniformly squeezed state,
we have
(2)
fmn '
Pmn
(1) (1)
Pm Pn
=3+
1
,
n̄
(9.97)
for any detuning ∆ = ±νp and any choice of ions (m, n) to measure. The value depends on
the squeezing parameter through n̄. The reason the two do not agree in the limit n̄ → 0
is that in that case Pm = Pn = 0, so fmn is undefined. The behavior of this function, too,
is consistent with the expected results for a second-order correlation function for such a
squeezed state [54].
9.7
Discussion and Conclusion
There are really two pieces of information that can be gleaned from the correlation function Pmn . The first is the structure of the normal modes. This structure can be probed by
detuning the interaction laser to the desired mode’s resonant frequency νp and reading out
ion m and n. Doing this for either the thermal state or the uniformly squeezed state from
the previous section give correlations as in Figure 9.2.
118
The other piece of information obtainable from Pmn is its relation to the single-ion detection probabilities Pm and Pn . This information is embodied in the normalized function fmn ,
and can be used to probe more general characteristics about the state. For instance, in the
comparison of squeezed and equivalent thermal states, with equivalence defined as an equal
average number of phonons in the mode being detected (i.e., n̄ = n̄p for ∆ = νp ), many of the
parameters cease to be important (detection time, coupling constant, geometric terms, etc.),
and information about the basic nature of the state—in this case, thermal or squeezed—is
revealed.
There is still much work to be done in exploring properties of correlations in a string
of trapped ions. The work presented here is designed to be a significant start in that
direction. We began with a definition of a measurable correlation function and single-ion
detection probabilities and connected these to two- and four-point functions of a general
quantum state, all in the perturbative regime. Next, we specialized to Gaussian states and
provided explicit formulas for these probabilities in terms of the state’s covariance matrix
elements. The examples of thermal and squeezed states were compared, and a normalized
correlation signature contrasted in each case, which shows agreement with standard results
for correlation functions used in quantum optics. Open problems include vast opportunities
to generalize these results to other interesting states, such number states, coherent states,
superpositions versus mixed states, and many others. The structure of a string of trapped
ions approximates a scalar field, but the deviations from this approximation could be a
source of interest, as well. Finally, a completely unexplored avenue would be to look at the
behavior of these correlation functions as the trap frequency is modulated, to see whether
such a modulation has a detectable correlation signature.
9.A
Appendix: Wick’s Theorem
Wick’s theorem theorem is often stated as a result for time-ordered expectation values of the
ground state of a multimode bosonic or fermionic system, as in quantum field theory [139],
but it can actually be used with any state and any prescribed ordering of the operators.
Wick’s theorem is commonly stated as [139]:
T {φm1 (t1 ) · · · φmN (tN )} = :φm1 (t1 ) · · · φmN (tN ): + :(all possible contractions): ,
(9.98)
where the colons in :(operators): place the contained operators in normal order (i.e., all
raising operators are to the left of all lowering operators), a “contraction” of two operators
is written as (and defined by)

[ψ (t ), ψn† (t )] if t > t ,
m j
j
k
k
φm (tj )φn (tk ) :=
†
[ψ (t ), ψm
(tj )] if tj < tk ,
n k
119
(9.99)
†
where ψm (t) and ψm
(t) are defined in Eq. (9.18), and “all possible contractions” means
every unique way of contracting pairs of operators together in this fashion.
We can make two generalizations: to anti-time-ordered products and to products with
no time ordering. Since normal ordering already prescribes and order for the uncontracted
operators, the only place time ordering shows up at all is in the contracted terms. As such,
we define a second type of contraction for this purpose:

[ψ (t ), ψn† (t )] if t < t ,
m j
j
k
k
φm (tj )φn (tk ) :=
†
[ψ (t ), ψm
(tj )] if tj > tk .
n k
(9.100)
Finally, if the operator order is not prescribed by either type of time ordering (and is to be
taken as given), then we shall define
φm (tj )φn (tk ) := [ψm (tj ), ψn† (tk )] .
(9.101)
Generalizing the usual inductive method of Ref. [139], it’s straighforward to generalize
Wick’s theorem to the following:
P{φm1 (t1 ) · · · φmN (tN )} = :φm1 (t1 ) · · · φmN (tN ): + :(all possible contractions): ,
(9.102)
where P{· · · } prescribes a time ordering for (some of) the operators within it, and the
contraction terms now must respect this ordering, using one of the above definitions in each
case according to the prescribed ordering of the two operators involved.
For our purposes, we are interested in the four-point function in Eq. (9.44). This also
serves as a particularly good but simple example of how to use Eq. (9.102). We’ll use the
following abbreviations to save space:
φn (t1 ) → φ1 ,
φm (t2 ) → φ2 ,
φm (t3 ) → φ3 ,
φn (t4 ) → φ4 .
(9.103)
We then can use Eq. (9.102) to make the following expansion:
T̄ {φ2 φ1 }T {φ3 φ4 } = :φ2 φ1 φ3 φ4 :
+ :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 :
+ :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 :
+ :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 : + :φ2 φ1 φ3 φ4 : .
(9.104)
Notice how the contraction type is determined by whether the terms are prescribed to be in
time order (φ3 and φ4 ), anti-time order (φ2 and φ1 ), or as written (all other combinations).
We need the expectation value of each of these terms. Since contractions of position opera-
120
tors are c-numbers, they can be taken out of the normal ordering and out of all expectation
values, giving
hT̄ {φ2 φ1 }T {φ3 φ4 }i = h:φ2 φ1 φ3 φ4 :i
+ φ2 φ3 h:φ1 φ4 :i + h:φ2 φ3 :iφ1 φ4 + φ2 φ3 φ1 φ4
+ φ2 φ4 h:φ1 φ3 :i + h:φ2 φ4 :iφ1 φ3 + φ2 φ4 φ1 φ3
+ φ2 φ1 h:φ3 φ4 :i + h:φ2 φ1 :iφ3 φ4 + φ2 φ1 φ3 φ4 .
(9.105)
The grouping of the terms in Eq. (9.105) suggests an alternate way of writing the expectation
value of this expression. Using Eq. (9.102), we have
hφ2 φ3 ihφ1 φ4 i = h:φ2 φ3 :ih:φ1 φ4 :i + φ2 φ3 h:φ1 φ4 :i + h:φ2 φ3 :iφ1 φ4 + φ2 φ3 φ1 φ4 ,
(9.106)
hφ2 φ4 ihφ1 φ3 i = h:φ2 φ4 :ih:φ1 φ3 :i + φ2 φ4 h:φ1 φ3 :i + h:φ2 φ4 :iφ1 φ3 + φ2 φ4 φ1 φ3 ,
(9.107)
hT̄ {φ2 φ1 }ihT {φ3 φ4 }i = h:φ2 φ1 :ih:φ3 φ4 :i + φ2 φ1 h:φ3 φ4 :i + h:φ2 φ1 :iφ3 φ4 + φ2 φ1 φ3 φ4 ,
(9.108)
Using this, we can write
hT̄ {φ2 φ1 }T {φ3 φ4 }i = hφ2 φ3 ihφ1 φ4 i + hφ2 φ4 ihφ1 φ3 i + hT̄ {φ2 φ1 }ihT {φ3 φ4 }i
+ h:φ2 φ1 φ3 φ4 :i − h:φ2 φ3 :ih:φ1 φ4 :i − h:φ2 φ4 :ih:φ1 φ3 :i − h:φ2 φ1 :ih:φ3 φ4 :i . (9.109)
Recalling the abbreviations (9.103), either Eq. (9.105) or Eq. (9.109) may be used to calculate Pmn to lowest nontrivial order.
9.A.1
Gaussian States
If the state has a Wigner function that is a Gaussian with zero mean, then all of the normalordered terms in Eq. (9.109) must cancel out in order to agree with the result known as
the “generalized Wick’s theorem” from Ref. [140]. As it is stated in Ref. [140], the theorem
applies only to ground-state expectation values of raising and lowering operators, but it can
be generalized to a much larger class of expectation values. Let κi ∈ {ap , a†p | p = 1, . . . , N }
be any raising or lowering operator. The generalized Wick’s theorem states that any groundstate expectation value of an even number of such operators can be written in the following
form:
h0| κ1 κ2 κ3 · · · κ2n |0i =
X
h0| κ1 κ2 |0i h0| κ3 κ4 |0i · · · h0| κ2n−1 κ2n |0i ,
Pd
121
(9.110)
where the sum is over all distinct permutations Pd of the 2n indices that preserve the
operator ordering within each pairing on the right—i.e., all permutations which give a
distinct product of expectation-value pairs hκl κm i and for which l < m in each of these
P
pairs. Also, any linear functions Fi = i Aij κj of the raising and lowering operators will
separate in a similar fashion:
h0| F1 F2 F3 · · · F2n |0i =
X
h0| F1 F2 |0i h0| F3 F4 |0i · · · h0| F2n−1 F2n |0i ,
(9.111)
Pd
as can be verified by direct calculation. Furthermore, it turns out that this property holds
for all zero-mean Gaussian states, both pure and mixed. To show this, one notes that any
such Gaussian state can be written as the partial trace of a pure Gaussian state of twice
as many modes [141]: ρ = trB |χihχ|. Such a Gaussian pure state |χi is unitarily related
to the vacuum state on the doubled mode set—i.e., |χi = Uχ |0, 0i. However, this unitary
transformation can be viewed in the Heisenberg picture as a symplectic linear transformation
P
on the (extended) vector of canonical operators [53], viz. Uχ† κj Uχ = k Lχjk ϕk =: Gj , where
κj is any raising or lowering operator in the original set, while ϕk is any such operator in
the extended set. From this, we can write expectation values as
hκ1 · · · κn i = tr(ρκ1 · · · κn )
= trA trB (|χihχ|)κ1 κ2 κ3 · · · κ2n
= hχ| κ1 · · · κn |χi
= h0, 0| Uχ† κ1 Uχ · · · Uχ† κn Uχ |0, 0i
= h0, 0| G1 · · · Gn |0, 0i .
(9.112)
Thus, we have
hκ1 κ2 κ3 · · · κ2n i = h0, 0| G1 G2 G3 · · · G2n |0, 0i
= h0, 0| G1 G2 |0, 0i h0, 0| G3 G4 |0, 0i · · · h0, 0| G2n−1 G2n |0, 0i
= hκ1 κ2 ihκ3 κ4 i · · · hκ2n−1 κ2n i ,
(9.113)
where we applied Eq. (9.111) in the middle step. Thus, the generalized Wick’s theorem holds
for all Gaussian states with zero mean. Furthermore, we may take linear combinations of
κj operators, and the theorem still holds:
hF1 F2 F3 · · · F2n i =
X
hF1 F2 ihF3 F4 i · · · hF2n−1 F2n i .
Pd
122
(9.114)
If the state in Eq. (9.109) is a zero-mean Gaussian, then Eq. (9.114) applies, resulting in
only the first three terms on the right being kept:
Gaussian
hT̄ {φm (t2 )φn (t1 )}T {φm (t3 )φn (t4 )}i −−−−−→ hφm (t2 )φm (t3 )ihφn (t1 )φn (t4 )i
state
+ hφm (t2 )φn (t4 )ihφn (t1 )φm (t3 )i + hT̄ {φm (t2 )φn (t1 )}ihT {φm (t3 )φn (t4 )}i . (9.115)
Comparing this with Eq. (9.39), we see that the first term will always give simply Pm Pn ,
and the second will always give a term similar to this, but with a different geometric factor.
Thus, we need only ever explicitly calculate the last two terms from Eq. (9.115).
We also note that if we restrict to Gaussian vibrational states, the generalized Wick’s
theorem says that all higher-order joint probabilities (those with more excited ions, like
Plmn , and/or higher-order calculations) will all decompose into integrals and sums of expectation values of the form hφm (t1 )φn (t2 )i and hT {φm (t1 )φn (t2 )}i for arbitrary m and
n. (Note that the antitime-ordered case is just the complex conjugate of the time-ordered
case). When restricting to Gaussian states, then, only these two types of expectation values
are required.
123
Part III
Entanglement in Curved Spacetime
124
Chapter 10
Relativistic Quantum Information
Theory
Nature is quantum mechanical. Nature is also relativistic. Neither of these properties is
intuitive to us as human beings because we experience the world at macroscopic, nonrelativistic scales. Still, much progress has been made in an information theoretic understanding
of nonrelativistic quantum theory—what is commonly called quantum information theory—
but this theory is not the best we can do, since we know that relativity must come into
the picture at some point. Curiously, limited progress has been made at this convergence
of these two disciplines, and a plethora of open problems remain [19], yet I believe it to be
one of the most important interfaces in modern physics (see Chapter 12). In this chapter,
I will guide a brief tour of some of the key results at the interface of quantum information
theory and relativity, stopping along the way for an in-depth look at the concepts necessary
to understand my work on entanglement in curved spacetime, described in Chapter 11.
10.1
The Fall of Local Realism
One of the most rudimentary places where relativity and quantum mechanics must be
considered together is with respect to the notion of entanglement. Up until now, I have
used this term to mean “having correlations stronger than any classical model would allow.”
Here I illustrate what this means and what it has to do with relativity.
10.1.1
Entanglement
All quantum states are described by a positive density operator ρ, which may be written as
a convex combination of pure states (i.e., rank-one projectors):
ρ=
X
ci |ψi i hψi | ,
i
125
(10.1)
where ci ≥ 0, and
P
i ci
= 1. This expansion is not unique, and the upper limit on the sum is
unspecified because different expansions can have different numbers of terms. The simplest
decomposition (i.e., the one with the smallest number of terms) is the eigendecomposition,
which always exists.
For an entangled state of a multi-particle system, these states |ψi i cannot be tensor
products of states of the individual constituent systems. Any state that has such a decomposition is called a separable state. An example of an entangled two-qubit state is the
singlet state
1 |ψe i = √ |01i − |10i ,
2
(10.2)
where |abi is understood to mean |ai ⊗ |bi, which has an associated density operator
ρe = |ψe i hψe | =
1
|01ih01| − |01ih10| − |10ih01| + |10ih10| .
2
(10.3)
We now attempt to write this density operator as
ρ=
XX
j
cjk |φj i hφj | ⊗ |φk i hφk | ,
(10.4)
k
where the sums run over a number of vectors |φn i that form a general overcomplete basis
(i.e., a basis with more vectors than necessary to span the space) for the space of one
qubit. The possible tensor products of these vectors then form an overcomplete basis for the
composite system. (The connection with Eq. (10.1) is made by writing |ψi i = |φj i⊗|φk i and
ci = cjk .) This decomposition is, in general, possible since we have defined the projectors
as forming an overcomplete basis for the 2-particle system, but for the case of ρe we find
that at least one cjk < 0 for any choice of tensor-product basis {|φj i ⊗ |φk i} [142], meaning
that this density operator represents a state of entanglement of the two constituent qubits.1
The most important difference between entangled and separable states is that because
separable states may be written as a convex combination of tensor-product states, the constituent qubits may be modelled as a collection of classical tops whose spin-axes are oriented
in individual, particular directions. The “true” configuration is determined stochastically,
but this stochasticity necessarily assumes that the system has a definite configuration at
all times, even though we do not know for sure what it is. Because of our ignorance of the
system’s “actual configuration,” all we can assign to the system are probabilities for the
various possible configurations, but it is always assumed that one of these configurations
truly represents the state of affairs in the system. The property described here is called local
1
Recall that it is always possible to write the density operator as a convex combination of a general basis
set as in Eq. (10.1). What makes a state entangled is that in such a representation, this basis can never be
a tensor-product basis. If we were to attempt expansion of an entangled state in any tensor-product basis,
as in Eq. (10.4), the expansion coefficients would not all be nonnegative.
126
realism. It is called realistic because each qubit can be modelled as a top whose spin-axis
actually points in a given direction; it is called local because the spin-axis of each qubit
exists independently of that of the other qubits, and thus measurements of the spin-axis
of one qubit do not affect measurements of the spin-axes of the other qubits. Correlations
may exist between these spin directions, but such correlations can be modeled as a classical average over the various choices of spin configurations, weighted by some probability
distribution that is given by the (nonnegative) coefficients cij in the decomposition of the
density operator (10.4). Any state that is separable has no entanglement.
Both separable and entangled states can display correlations in their measurement statistics. The difference is in the assumptions that are allowed in each case. The (entangled)
singlet state ρe above has the property that measurements made in the same basis for both
qubits always display anti-correlation; that is, regardless of the basis chosen, one qubit will
always be measured in the “up” state, while the other is in the “down” state. This type of
correlation as stated here doesn’t seem all that surprising because it is easily confused with
the following completely classical case.
Consider a pair of classical tops that are prepared (by some device or trustworthy person)
with anti-correlated axes of rotation. Specifically, the axis of the first top is uniformly
random on its own but is guaranteed to point opposite to that of the other top, which is also
uniformly random when considered alone. Once I observe the axis of rotation for one of the
tops, I can immediately infer that the other top is rotating in the opposite direction about
the same axis (see Fig. 10.1). This picture makes no unacceptable demands of us with regard
to how the world works. Each top has an orientation; we just don’t know what it is. As such,
we use a probability distribution to describe the configuration, and this distribution entails
correlations in the orientation of the tops. The connection with a quantum mechanical
singlet state is a spurious one, however, as the next section demonstrates: one cannot think
of the two qubits as each having a definite pre-measurement spin orientation—an orientation
that was revealed (as up or down along some axis) through the measurement process—
without allowing for faster-than-light propagation of information. Quantum theory hath
slain local realism.
10.1.2
Bell Inequalities and Hidden-Variable Models
The property of local realism is the statement that subsystems of a composite system each
have real properties that can be measured independently (realism), without such measurements affecting or depending on other subsystems (locality). The assumption of realism is
not at all unreasonable, considering it is true of most of our everyday experiences. In fact,
it is ingrained at such a low level in our consciousness that it usually takes some work to
even realize that the assumption is being made. The motivation for locality, however, is
based on one of the pillars of relativity: the speed of light is a constant, which implies that
nothing can travel faster than light. Allowing a measurement at one spacetime location to
127
inferred
observed
Figure 10.1: Spin axis anti-correlation for two classical tops. Each individual top has an axis
of rotation. This axis can point in any direction, uniformly randomly, but with the restriction
that the spin axes of both tops are perfectly anti-correlated. That is, if an observation of the
first top reveals that the red vector is the spin axis, then the second top is known to be spinning
along its (opposite-pointing) red vector, as well. The same would be true if the result were
the green or purple vector instead. This type of correlation is strong (i.e., knowledge of one
determines the other completely), but it still admits a local realistic description because each top
has a preexisting spin (red, green, purple, etc.); we just don’t know what it is. This preexisting
property of the top is revealed upon observation (realism). Thus, no superluminal signaling
is needed to “update” the spin of the other top after observing the first one (locality)—they
were in this configuration the whole time, even though we didn’t know it. Perhaps surprisingly,
the entangled state of Eq. (10.2), a singlet, does not admit this (or any other) local realistic
description (see Section 10.1.2).
128
be instantaneously affected by what’s happening at another spacetime location (for example, another measurement) would, in the case of spacelike separation, require propagation
of some sort of signal faster than light in order to update “the real state of affairs” at the
distant location.
Einstein, Podolsky, and Rosen (EPR) claimed in 1935 that quantum mechanics, in
its probabilistic wave-function description, is incomplete [33] because it does not separately
account for the apparently real properties of two spatially separated particles if such particles
are in an entangled state. Bohr’s reply to this assertion followed quickly [143] in an attempt
to argue that the wavefunction is all that is physically predictable for a system. Nonetheless,
Einstein held fast to a belief in a local realistic description of nature that retains the intuitive
notions that all composite systems whose components are spatially separated should admit a
real state for each component that can be measured independently of the other components.
Einstein asserted that if the wavefunction in some cases cannot describe the real state of one
subsystem independently of the real state of another, then the wave-function description of
nature must be incomplete and should be supplemented with other variables. This belief in
so-called “hidden variables” remained at least plausible until 1964 when Bell formulated his
famous inequality [11], which ruled out a local realistic description of entangled states that
is consistent with the statistical predictions of quantum mechanics. Subsequent experiments
consistently supported quantum mechanics, thus ruling out any local realistic description of
entangled states [144]. What follows is a description of Bell’s original inequality [11]. (See
also Refs. [144, 145, 146] for more information and examples of other Bell inequalities.)
We begin by positing a system of two qubits that are in the (entangled) singlet state of
Eq. (10.2), repeated here for reference:
1 |ψe i = √ |01i − |10i ,
2
with ρe = |ψe ihψe |. A measurement of the spins of the two constituent qubits along any
single axis a—i.e., σ · a ⊗ 1 for the first qubit and 1 ⊗ σ · a for the second qubit, where
σ is a vector of Pauli matrices (σx , σy , σz )—will result in perfect anticorrelation of the
measurement results (i.e., one will be up, while the other will be down).2 Furthermore, we
may define the correlation function,
CQM (a, b) := hσ · a ⊗ σ · bi = tr(ρe σ · a ⊗ σ · b) = −a · b ,
(10.5)
for arbitrary choices of spin directions, a for the first qubit and b for the second qubit (note
that C(a, a) = −1, as required). Eq. (10.5) is the prediction made by quantum mechanics
for the correlation of measurement results along arbitrary axes of a two-qubit system in the
(entangled) singlet state (10.2).
2
Notice the apparent similarity with Fig. 10.1. Dispelling this spurious connection is the primary objective
of this section.
129
Motivated by a desire to have noncausal measurement processes, we wish to model this
correlation function in terms of two deterministic measurement functions A and B, which
are each functions only of the spin-measurement direction a or b for the associated qubit,
plus a general hidden variable λ. This hidden variable may be any mathematical object
or collection of such objects (scalars, vectors, tensors, functions, matrices, etc.), but we
will treat it formally as if it were a continuous single parameter. If we wish to model the
behavior of a quantum system in terms of these measurement functions, then the functions
can only give out values of ±1; thus,
A(a, λ) = ±1 ,
B(b, λ) = ±1 .
(10.6)
It is through this way of writing of the measurement functions that locality is enforced: the
measurement of the spin of qubit 1 does not depend on the direction of the measurement
being made on qubit 2, and vice versa. If we had instead written the measurement function
for the first qubit as A(a, b, λ), then we would be constructing a nonlocal model. Although
we cannot know the exact value of λ (which is why it is called a hidden variable), it is
assumed to have a definite value, which is known probabilistically, and it is this probability
distribution P (λ) for the hidden variable that results in different “states” of the system.
The expectation value of the correlated measurements of A and B is then given by
Z
CHV (a, b) =
dλ P (λ)A(a, λ)B(b, λ) ,
(10.7)
where the “HV” subscript indicates that this is the correlation predicted by the hiddenvariable model. By requiring that P (λ) be a nonnegative, normalized probability distribution over the possible actual values of λ, we enforce our second requirement, realism. The
nonnegativity of this distribution implies that the hidden variable λ actually has one of its
possible values; we don’t know which one exactly, so we use a probability to describe our
level of surety for each possible value of being the actual value.
Presuming quantum mechanics to give correct statistical predictions, the above expectation value must equal the quantum mechanically predicted correlation function for this
system, i.e.,
CHV (a, b) = CQM (a, b) = −a · b
(10.8)
for all choices of measurement directions, a and b. It will be shown presently that this is, in
general, impossible. Contradiction is demonstrated through the example of the singlet state
given by Eq. (10.2). The requirement that CHV (a, a) = −1 for agreement with quantum
mechanics and the restrictions placed on P (λ) require that
A(a, λ) = −B(a, λ)
130
(10.9)
for all choices of a. Thus, we may rewrite Eq. (10.7) as
Z
CHV (a, b) = −
dλ P (λ)A(a, λ)A(b, λ) .
(10.10)
If c is another measurement direction, then it follows from this result that
Z
CHV (a, b) − CHV (a, c) = − dλ P (λ)[A(a, λ)A(b, λ) − A(a, λ)A(c, λ)]
Z
= dλ P (λ)A(a, λ)[A(c, λ) − A(b, λ)]
Z
= dλ P (λ)A(a, λ)[A(b, λ)A(b, λ)A(c, λ) − A(b, λ)]
Z
= dλ P (λ)A(a, λ)A(b, λ)[A(b, λ)A(c, λ) − 1] ,
(10.11)
where the next-to-last equality follows from Eq. (10.6), from which we obtain
Z
|CHV (a, b) − CHV (a, c)| ≤
dλ P (λ)[1 − A(b, λ)A(c, λ)] .
(10.12)
The right-hand side is just 1 + CHV (b, c), which implies that
1 + CHV (b, c) ≥ |CHV (a, b) − CHV (a, c)| .
(10.13)
It is straightforward to see that this inequality is violated by the quantum mechanical
correlation function (10.5). If we choose our unit vectors a and b to be perpendicular, and
we choose c to bisect the angle made by a and b, then CHV (a, b) = 0, and CHV (b, c) =
√
CHV (a, c) = − 2/2 ≈ −.707. In this case the inequality of Eq. (10.13) is violated because
1 − .707 = .203 6≥ .707 .
(10.14)
Thus, the singlet state of Eq. (10.2) cannot be modeled in a local realistic manner. In
particular, measurements on a singlet cannot be represented by statistics derived from the
model in Fig. 10.1, nor from any other model that involves the intuitive concepts of locality
and realism.
10.1.3
Implications
The inequality in Eq. (10.13) is called a Bell inequality after its creator, John Bell [11].
This inequality applies specifically to the (entangled) singlet state of two qubits [147], but
all inequalities of this sort—i.e., those that place restrictions on correlations established
through a local realistic hidden-variable model—are also called “Bell inequalities.” Notice
that in no way have we restricted what the hidden variable λ is. Thus, this result applies
to all hidden variable models for a singlet state that may be formulated with a correlation
131
function as in Eq. (10.7) with the further requirements that P (λ) ≥ 0 and
R
dλP (λ) = 1.
Models of this sort automatically satisfy the requirements of locality through the fact that
A(a, λ) 6= A(a, b, λ) (and similarly for B) and realism through the restriction that P (λ) be
a true probability distribution. Because Bell inequalities are founded on the assumptions
of locality and realism, any system that admits a local realistic hidden-variable model
description automatically satisfies any Bell inequality that may be derived for that system.3
What has been shown here is that there exist entangled states in quantum mechanics
whose measurement statistics violate the concurrent assumptions of locality and realism.
What has not been shown is that all entangled states fail to admit a local realistic description. In fact, such entangled states exist [148, 149]. Nevertheless, it has been shown
recently that all bipartite entangled states display some property that cannot be modeled
in a local realistic framework [12]. This brings down the gavel on the question of the dividing line between classical and quantum correlations in the bipartite case: the presence of
entanglement marks the division.
With the concurrent assumptions of locality and realism rendered untenable in quantum
mechanics, where do we go from here? The two obvious courses of action are (1) to discard
locality, and (2) to discard realism. Despite its refusal to submit to our classical intuition,
entanglement cannot be used for superluminal signaling [19], which keeps both prospects
very much on the table, for even if ontological locality (i.e., locality at all levels of reality)
is abandoned, causality for all practical purposes is still preserved.
All interpretations of quantum mechanics based on nonlocal hidden variables by definition choose the first option. In many ways, it seems like the “easier” one, since it may very
well be that nature, at its lowest levels, is nonlocal, even though we as macroscopic human
beings will never be able to see it. Relativity, after all, was formulated before quantum mechanics, and so it could be that the universal speed limit it imposes is an effective concept,
not a primary one at the ontological level. The de Broglie-Bohm interpretation of quantum mechanics is currently the most well-studied example [16, 17, 18]. In this model, the
wavefunction acquires ontological status in the form of a “pilot wave” that pushes particles
around. When measurements are made, the particles must snap into place instantaneously
in order to satisfy the new post-measurement wavefunction, thus violating relativity at
the ontological level. Making predications consistent with standard quantum mechanics,
superluminal signaling is still prohibited.
Discarding realism requires a much larger leap of understanding. The belief that objects
have real properties, whether they can be measured or not, is so ingrained within us by our
experience that leaving it behind often poses a challenge. Some might be tempted to view
abandoning this view as tantamount to solipsism—the idea that oneself is the only this that
really exists. Discarding realism is not solipsistic, however; it merely requires understanding
3
I often wonder what Einstein, a staunch realist [33], would have thought about Bell’s result [11] had be
lived long enough to see it.
132
that the existence of an objective, external world does not necessarily require the idea of
fixed properties of the constituent objects within it. In fact, it need not require objects at
all! The idea of an object could be seen as an effective concept rather than an ontological
entity. Properties of objects would be of the same character.
Several avenues proceed in this direction. The first is the standard Copenhagen interpretation, taught in most modern textbooks on quantum theory, even though it is not well
defined and often interpreted differently by different authors [150]. It involves a definite
abandonment of realism, however, because measurements are not assumed to reveal preexisting properties. Another is the relational interpretation, most often associated with Carlo
Rovelli [151]. In this picture, a quantum state only has meaning relative to an observer.
Thus, properties only exist relative to an observer, as well. A third interpretation that
arguably goes the furthest in rejecting realism is the Bayesian interpretation, promoted
chiefly by Carl Caves, Chris Fuchs, and Rüdiger Schack [152], which also is arguably the
most commensurate with quantum information theory [153]. The Bayesian interpretation
treats quantum states as representing an agent’s state of belief about future measurement
outcomes. In this way, the state becomes like a probability distribution in the Bayesian
sense—that is, it represents the odds that an agent would assign to a possible gambling
enterprise based on the event in question. This correspondence entails one important difference: in the classical Bayesian context, a probability distribution is used only to quantify
ignorance about the “real state of affairs”—that is, realism is an allowed (and often tacitly
implied) assumption. The quantum Bayesian perspective assumes locality and thus prohibits this luxury. In fact, the luxury is extraneous, if convenient, in the classical case, since
it may not be the case that something is decided beforehand. Quantification of ignorance
of a future measurement outcome as a probability distribution or a quantum state, in each
respective case, doesn’t require a world-view based on realism.
It should be clear by now that the Bayesian interpretation is the view that I personally
favor at this time. Still, this is intended to be a very biased and not-all-inclusive list whose
purpose is to illustrate the fracturing that has occurred in the physics community regarding
the foundations of quantum mechanics. The implications of Bell’s inequality [11] have not
yet settled out, and quantum foundations remains a widely pursued endeavor and, I believe,
an important open problem. This issue will be revisited briefly in Chapter 12. For now,
I wish to move on to a discussion of quantum field theory in curved spacetime and its
implications for quantum information theory.
10.2
Quantum Information Theory in Curved Spacetime
Quantum theory in the full context of special relativity requires the use of quantum fields [139].
Quantum field theory in Minkowski spacetime is well studied and is the foundation of all
modern particle physics. As such, the reader is encouraged to peruse any of the available
133
comprehensive textbooks on the subject for an introduction to quantum fields, including
Peskin and Schroeder’s An Introduction to Quantum Field Theory [139], which was my main
reference. This section will focus on extending quantum field theory to curved spacetime,
a first step in the direction of incorporating general relativity in a quantum context, while
focussing on the information-theoretic implications of this generalization. Most of the discussion here will follow Quantum Field Theory in Curved Space by Birrell and Davies [154]
and the recent review article by Peres and Terno entitled “Quantum information and relativity theory” [19], to which I encouraged the reader to refer for additional details.
10.2.1
Particles? What particles?
The notion of a particle is unambiguous in Minkowski-space quantum field theory.4 A
particle can be identified with an excitation of a field quantum |1i = a† |0i, where a† is
the creation operator for some field mode, and |ni is a number state for that mode, with
|0i indicating the vacuum. The tacit assumption in this definition, though, is that the
vacuum state is unique. In Minkowski-space quantum field theory, it is true that all inertial
observers agree on the vacuum state, but this does not generalize to accelerated observers
in Minkowski spacetime. When spacetime curvature is present, even inertial observers may
disagree on what state corresponds to “the vacuum.” That is to say, one observer may
detect particles based on her trajectory through the “vacuum state” of a different observer,
who registers no particles in his detectors. The fact that this doesn’t happen for inertial
observers in Minkowski space is due to Poincaré invariance, which is respected by the set
of inertial observers and of the vacuum state definition
ak |0i = 0
for all modes k.
(10.15)
In general, such symmetries are absent in curved spacetimes, so “the vacuum” becomes an
observer-dependent concept. From an ambiguous vacuum state, the notion of “particles”
inherits this ambiguity. The ambiguity is related to the trajectories of individual observers,
though, so it can be dealt with properly by defining a “particle” as that which is detected by
a (local) particle detector, carried along by an observer [123, 154, 155, 156]. This operational
approach renders moot the debate about what a particle actually is by recognizing that the
question requires an observer-dependent answer.
10.2.2
Introducing Curvature
While the notion of particles in curved-space quantum field theory is more subtle than
in the Minkowski-space version, some properties carry over in an easily generalized way.
Placing quantum field theory on a curved background is an important first step toward
incorporating general relativity into the quantum framework. The key difference is that the
4
Well, it’s an unambiguous as it’s going to get in a quantum context.
134
background spacetime is not quantized and instead evolves kinematically and classically.
Often, direct appeal to the Einstein equations is unnecessary, since the spacetime is taken
to have an imposed (but still useful) structure. My work in the next chapter follows this
method, imposing a de Sitter (exponentially expanding) spacetime [157, 158], known to be
useful in inflationary cosmological models [159].
Imposing curvature in quantum field theory, in this semi-classical fashion, is much the
same as in classical theories: the Minkowski metric ηµν becomes a pseudo-Riemannian metric gµν , and Lagrangians and equations of motion written in covariant (coordinate-free) form
are promoted without formal modification to their curved-space counterparts. On occasion,
extra curvature terms (which would be absent in Minkowski spacetime) supplement this
simple curved-space promotion procedure (see Ref. [154] for more details).
The work in the next chapter considers a scalar field, which will serve as an illustrative
example of the simplicity of this method. For a general metric given by5
ds2 = gµν dxµ dxν ,
(10.16)
the scalar field Lagrangian density is written as [154]
L(x) =
h
i
1p
−g(x) ∇µ φ(x)∇ν φ(x) − m2 φ2 (x) − ξR(x)φ2 (x) ,
2
(10.17)
where φ(x) is the scalar field operator at spacetime location x, g(x) = det gµν (x), ∇µ is the
covariant derivative with respect to xµ , m is the mass of the field, R(x) is the Ricci scalar
for the background spacetime (a measure of curvature), and ξ is a numerical factor. The
last term ξRφ2 is included as the only possible local, scalar coupling to the metric that
has the right dimensions. It vanishes in the Minkowski-space case and is therefore a valid
possible generalization from that case. This Lagrangian leads to the following equation of
motion for the scalar field:
µν
g ∇µ ∇ν + m2 + ξR(x) φ(x) = 0 ,
(10.18)
which again resembles the Minkowski-space counterpart with the prescription that ηµν →
gµν , ∂µ → ∇µ , and ξRφ2 is included (for some value of ξ). The statistics of detected φparticles is discussed in the next chapter, so I will not elaborate on it here. (More details can
be found in Ref. [154].) The main message at this point is that general covariance makes it
easy to generalize Minkowski-space quantum field theory calculations to curved-space ones,
although the results—as we shall see presently—are very different.
10.2.3
5
Horizons, Radiation, and Quantum Information Theory
The Einstein summation convention is implied unless otherwise stated.
135
The ambiguity in the vacuum discussed in Sec. 10.2.1 leads to a host of interesting particle
creation phenomena, perceived by the detector in question as a bath of radiation. The first
case to be considered here is that of Unruh-Davies radiation [155, 156]. This is the radiation
that results from the nonequivalence of the Rindler vacuum (that perceived by a uniformly
accelerating observer) and the Minkowski vacuum (that perceived by inertial observers) in
Minkowski spacetime. This radiation is detected when a uniformly accelerating observer
travels through the Minkowski vacuum. He perceives a temperature in the universe of
TUD =
a
,
2π
(10.19)
where a is his acceleration, and natural units are employed. The second type of radiation is
Hawking radiation [160], which is radiation from the horizon of black holes. Such radiation
is emitted from a Schwarzschild black hole at a temperature of
TH =
κ
,
2π
(10.20)
where κ = 1/4M is the surface gravity of a Schwarzschild black hole with mass M . The final
effect in this class is Gibbons-Hawking radiation [123], which is radiation from an expanding
(de Sitter) universe [157, 158]. The perceived temperature by an inertial observer in this
universe is
TGH =
κ
,
2π
(10.21)
where κ is now the rate of (exponential) expansion (details in the next chapter).
If for no other reason than the obvious similarity of these formulas, it would seem that
these phenomena might be connected. While many interpretations are possible, the connection can be understood as due to the presence of a horizon6 in all three cases [19], allowing
virtual particles created at the horizon to escape to infinity if their partners fall behind it.
It is beyond the scope of my purpose here to delve in depth into the mechanism and consequences of horizon-generated radiation, although it is important to mention that horizons
present an unresolved theoretical and philosophical challenge in terms of what happens to
information that falls behind a horizon. This is especially vivid in the case of a black hole,
which eventually evaporates due to its emitted radiation [160], thus leaving the possibility
that infalling information is lost forever and that unitarity (in-principle reversibility) is not
a property of nature. This is known known as the black hole information paradox, and there
are currently a number of possible resolutions [19], some of which preserve the information
in fluctuations in the spectrum of emitted Hawking radiation or send it into alternate universes, while others suggest that non-unitary evolution is an intrinsic property of nature.
Gibbons-Hawking radiation will be discussed in more depth in the next chapter, while the
6
A uniformly accelerating observer has future and past horizons defined by the asymptotes of his hyperbolic trajectory through spacetime [19, 154].
136
reader is invited to consult Refs. [19] and [154] and references therein for more details on
all three types of radiation and on the black hole information paradox.
Spacetime curvature clearly changes the rules for phenomena in quantum field theory. While some concepts such as locality [139], are preserved, one might expect that the
radiation emanating from a horizon would generate decoherence and/or reduce entanglement. In Minkowski spacetime, entanglement is a Lorentz-invariant quantity, even if it gets
shuffled between spin and momentum due to the Wigner rotation of a boost [19, 161]. However, acceleration degrades entanglement [162] and reduces its capacity for use in quantum
information-theoretic applications, such as quantum teleportation [40, 163]. Deutsch [164]
and Bacon [165] showed that closed timelike curves, if they exist, might be used to implement nonunitary quantum evolution. This concept was generalized to a proposal by
Ralph [166, 167, 168] that uses a gravitational field to “slow the clock” of one of a pair of entangled particles, which can be used as a teleportation resource to implement the same types
of evolutions. Weak curvature was also considered by Ralph, Milburn, and Downes [169] in
its effect on entangled photons, leading to the possibility of gravitationally induced decoherence if one photon of an entangled pair experiences a changing gravitational field while
the other does not. Many of these ideas are controversial, and understandably, very little
experimental progress has been made on them. The purpose of listing them here is to
give a flavor for what kind of research avenues are available at the interface of quantum
informaiton theory and relativity (see also Ref. [19]).
The next chapter builds on a proposal by Reznik [170] and unites the two themes in this
chapter: violation of local realism and quantum field theory in curved spacetime. Reznik
proposes in Ref. [170] that geometric entanglement in the vacuum state of a Minkowskispace quantum field can be swapped to local two-state systems by a simple coherent coupling mechanism. Even though the entanglement thus generated is small for well-separated
systems, it can still be shown to violate a Bell inequality. I generalize this proposal by
comparing two modifications to it: (1) “heat the universe” to a thermal state with small
temperature (still in a Minkowski spacetime), and (2) perform no such heating but instead
allow the universe to expand at a very slow, but exponential, rate (now a de Sitter universe). By the Gibbons-Hawking effect [123], de Sitter expansion will appear to a local
detector in the same way that heating of a Minkowski universe would, and the temperatures in cases (1) and (2) can be chosen to be equal. Despite the local equivalence of the
two cases, entanglement—a quantum information-theoretic concept and resource—can be
used to distinguish curvature from heating [7].
137
Chapter 11
Entangling Power of an Expanding
Universe
This chapter describes the work presented in Ref. [7].
11.1
Introduction
Information in curved spacetime has played a prominent role in the attempt to understand
the interface between quantum physics and gravity [123, 160, 171, 172]. While abstract
properties of curved-space quantum fields (including their entanglement) can be studied directly [157, 173, 174, 175, 176], an operational approach involving observers with detectors
historically has been a critical component of theoretical progress in this area [123, 154]. With
the birth of quantum information theory [22], quantum systems could now be analyzed in
terms of their use for information-theoretic tasks like quantum computation [22], quantum
teleportation [40], and quantum cryptography [177]. Entanglement is a phenomenon that
is uniquely quantum mechanical in nature [12] and can be considered both an informationtheoretic and a physical resource [35]. It is known that the Minkowski vacuum possesses
long-range entanglement [19, 175] that can be swapped to local inertial systems using standard quantum coupling mechanisms [170]. Variations on this theme can be considered,
including accelerating detectors [178], thermal states [179], and curved spacetime. Our
focus will be on curvature. For this, we choose an exponentially expanding (de Sitter)
universe [157, 158] for its simplicity and because of its importance to cosmology [159].
We wish to demonstrate a connection between a physical property of spacetime (curvature) and an information-theoretic resource (entanglement). While it is possible to directly
study the entanglement present in a quantum field in de Sitter spacetime, this sometimes
leads to difficulties [174] that are not present in a more operational approach. Still, it is
known that entanglement between field modes can directly encode a spacetime’s curvature
parameters [176]. Motivated by a desire to be as operational as possible, we examine how
138
curvature affects a field’s usefulness as an entangling resource—i.e., its ability to entangle
distant quantum systems (“detectors”) using purely local interactions. We begin by reviewing the response of a single, inertial detector interacting with a massless, conformally
coupled scalar field. The result in the vacuum de Sitter case is identical to that in the case of
a thermal ensemble of field particles in flat spacetime [123, 154]. Next, we ask the question,
can entanglement be used to distinguish de Sitter vacuum expansion from Minkowski-space
heating? We show that with two detectors on comoving trajectories, there exists a parameter regime in which the local systems that couple to the field will become entangled despite
the presence of extra thermal noise in each individual detector. Interestingly, this region of
parameter-space in the expanding case is a proper subset of the same region in the locally
equivalent thermal case. Thus, while both universes affect a local inertial detector in exactly
the same way, entanglement between two detectors can be used to distinguish them.
11.2
Two Universes
We start with the following experimental setup, which is nearly identical to that used by
Reznik et al. [170], using units where ~ = c = kB = 1. We pose our problem completely in
operational terms, but our goal is to show proof of principle—not necessarily practicality of
the method. We suppose that the inhabitants of a particular planet launch a satellite into
space to measure the temperature of the universe they inhabit. On board this satellite is a
qubit (a two-level quantum system), initially in the ground state |0i, that gets coupled locally and for a limited time to a scalar field using a simple De Witt monopole coupling [105].
The time-dependent interaction Hamiltonian for this detector is, in the interaction picture,
HI (τ ) = η(τ )φ x(τ ) e+iΩτ σ + + e−iΩτ σ − ,
(11.1)
where τ is the proper time of the satellite, η(τ ) is a weak time-dependent coupling parameter
(which we’ll call the detector’s “window function”), x(τ ) is the worldline of the satellite, φ(x)
is the field operator at the spacetime location x, and the rest represents the interactionpicture Pauli operator σx (τ ) for the local qubit with (tunable) energy gap Ω. Roughly
speaking, the detector works by inducing oscillations between the two levels at a strength
governed by the local value of the field.
From now on, we refer to this qubit as a “detector,” although the process of “detection”
includes only the field interaction (before projective measurement). We wish to examine
when two such detectors become entangled through their local interactions with the field,
so we delay classical readout to allow for general quantum postprocessing, which may be
necessary to show violation of a Bell inequality [180].
The window function η(τ ) is used to turn the detector on and off, but the transitions
must be sufficiently smooth so as not to excite the field too much in the process [181].
Beyond this requirement, on physical grounds, our results should not depend on the details
139
of the window function as long as it is approximately time bounded, so we will always choose
η(τ ) to be proportional to a Gaussian,
η(τ ) = η0 e−(τ −τ0 )
2 /2σ 2
,
(11.2)
where η0 = η(τ0 ) 1 is a small unitless constant that enforces the weak-coupling limit
and allows us to use perturbation theory. This window function approximates the detector
being “on” when |τ − τ0 | . σ and “off” the rest of the time and also has a nice analytic
form.
Without loss of generality, we can set τ0 = 0. To lowest nontrivial order in η0 , the qubit
after the interaction (but before readout) will be found in the state
ρ = A |1ih1| + (1 − A) |0ih0| ,
(11.3)
where
Z
∞
A=
Z
∞
dτ
−∞
0
dτ 0 η(τ )η(τ 0 )e−iΩ(τ −τ ) D+ x(τ ); x(τ 0 ) ,
(11.4)
−∞
and where D+ (x; x0 ) = hφ(x)φ(x0 )i is the Wightman function for the field, with expectation
taken with respect to the state of the field (assumed to be a zero-mean Gaussian state, but
not necessarily the vacuum). Repeated measurement in the {|0i , |1i} basis for a variety
of values of Ω allows for determination of the state of the detector as a function of Ω.1
As is clear from Eq. (11.4), the state is completely determined by the detector response
function D+ x(τ ); x(τ 0 ) , which is the Wightman function taken at two different proper
times along the worldline of the detector [154].
We consider two possible universes. The first is Minkowski,
ds2 = dt2 −
3
X
dx2i ,
(11.5)
i=1
with the field in a thermal state with temperature T with respect to the inertial trajectory
{xi } = (constant). The second is a de Sitter universe,
2
2
2κt
ds = dt − e
3
X
dx2i ,
(11.6)
i=1
where κ is the expansion rate, in the conformal vacuum. The variables {xi } are comoving
coordinates, and t is cosmic time. (Since the Minkowski metric is the special case κ = 0,
this terminology carries over to it, as well.) In both universes, worldlines of constant {xi }
1
More general measurements will be required to demonstrate entanglement between two such detectors,
though.
140
are inertial trajectories (geodesics), and intervals of proper time equal those of cosmic time
(∆τ = ∆t). In both cases, the scalar field φ(x) is massless and conformally coupled [154],
satisfying
[x + 16 R(x)]φ(x) = 0 ,
(11.7)
where the Ricci scalar R(x) = 12κ2 is a constant, thus acting as an effective mass term,
proportional to the expansion rate κ.
Gibbons and Hawking [123] showed that the detector response function for any inertial
observer in the de Sitter case is exactly the same as that of a detector at rest in a thermal
bath of field particles with temperature T = κ/2π in flat spacetime. Thus, a single detector
alone cannot distinguish between the two cases if it forever remains on a given inertial
trajectory. In both cases considered above, the detector is at rest in the comoving frame
and thus [154],
T2
DT+ x(τ ); x(τ 0 ) = −
csch2 [πT (t − t0 − i)] ,
4
(11.8)
where the subscript T indicates that this is a detector response function for a thermal
state at temperature T . When the satellite begins sending back measurement data, the
reconstructed A(Ω) is found to be consistent with the detector being at rest in a thermal
bath of field particles at a small but nonzero temperature T . If the inhabitants wish to
know whether this perceived thermality is a result of heating or expansion, though, they
must be more creative.
Obviously, they could use astrophysical clues (like we have done on Earth) and/or
Doppler-shift measurements2 to determine whether their universe is expanding or not, but
we are going to restrict them to using only satellite-mounted detectors of the sort described
above on fixed inertial trajectories. If the detectors are to be useful, then, they will need
more than one.
11.3
Entanglement Signature
We propose the following alternative that makes use of entanglement to distinguish the two
universes. We imagine two satellites, each having many qubits that interact locally with
the scalar field. (Having many detectors allows access to many copies of the same state.)
We assume that the satellites have no initial entanglement with each other and that the
qubits each begin in the ground state. After interacting with the field, measurement is
delayed to allow for general quantum operations (local to each satellite) on the multitude
of qubits on board. In the end, however, the only data that can be transmitted back to the
2
The thermal Minkowski case exhibits Doppler shifting for detectors at different velocities [154]; the
vacuum de Sitter case does not [123].
141
home planet are measurement results, plus information about the postprocessing and the
particular measurements performed.
In an attempt to be as simple as possible, we analyze the case of two inertial detectors, a
and b, on the comoving trajectories x1 = ±L/2 (with x2 = x3 = 0). Due to the homogeneity
and isotropy of space in both scenarios, this case is remarkably general—but not entirely
so since one could imagine the detectors in motion with respect to each other (beyond the
relative motion generated by any expansion). For simplicity, we’ll also require that the two
detectors have synchronized local clocks with τa,b = t, equal resonant frequencies Ωa,b = Ω,
and identical window functions ηa,b (τ ) = η0 e−τ
2 /2σ 2
. Finally, we desire that L σ so that
the detector-field interactions can be considered noncausal events.3 As we shall see, these
restrictions will still allow the inhabitants, located at xi = 0, to distinguish expansion from
heating.
By spatial symmetry, each detector alone must respond using the detector response
function from Eq. (11.8) and thus provides no useful information. The only hope, then, is
in the correlations between the detectors. We will focus on those correlations that signal
the presence of entanglement of the detectors after interaction with the field. For a pair of
qubits, the negativity [182] of a state is nonzero if and only if the systems are entangled [183].
Since we have access to (by assumption) multiple copies of an entangled state of pairs of
qubits, a local measurement protocol (on the many copies of the state) always exists to
verify entanglement by showing a violation of a Bell inequality [180, 184]. This can be
verified by a third party using classical data received from both satellites.
We will focus on finding the regimes in which entanglement is nonzero, rather than on
the magnitude of the entanglement for two reasons. First, the amount of extractable entanglement is small enough to be impractical as a resource and will depend on the details of the
detector coupling. Second, we are primarily interested in understanding a qualitative difference between the quantum behavior of curved and flat spacetime; examining entanglement
ensures that this is a genuinely quantum mechanical effect [12].
An analogous calculation to Reznik’s [170] shows that the negativity of the joint state
of the qubits is
N = max |X| − A, 0 ,
(11.9)
where A is the individual detector response from Eq. (11.4), while X is defined as
Z
∞
X=−
Z
t
dt
−∞
Z
= −2
t0 <t
0 dt0 η(t)η(t0 )eiΩ(t+t ) D+ xa (t); xb (t0 ) + D+ xb (t); xa (t0 )
−∞
0
dt dt0 η(t)η(t0 )eiΩ(t+t ) D+ xa (t); xb (t0 ) .
3
(11.10)
Although the Gaussian window functions technically have tails that extend forever, none of the results
change if we assume a smooth cutoff of the Gaussian (to zero) around, say, 10σ as long as both L and T −1
are still much larger than this.
142
The limits of integration enforce time ordering [139], so we can use the Wightman function
(instead of the Feynman propagator) as shown. This is useful because symmetry of the
two detectors means that D+ xa (t); xb (t0 ) = D+ xb (t); xa (t0 ) , a fact used to obtain the
second line. This integral measures the amplitude that the detectors will exchange a virtual
particle, while A measures the probability that each detector becomes excited either by
absorbing or emitting a particle.
We begin by considering when the qubits become entangled when T = 0. (We also
define κ ≡ 2πT from now on so we can talk about expansion rates in terms of the associated Gibbons-Hawking temperature.) This case corresponds to the one considered by
Reznik [170] using different window functions. In the T = 0 case, the Wightman function
used in X is
D0+ xa (t); xb (t0 ) =
4π 2
−1
,
0
(t − t − i)2 − L2
(11.11)
and the detector response function (used in A) is obtained by letting L → 0 and is also
obtainable as the limit of Eq. (11.8) as T → 0. Both X and A can be evaluated analytically:
L2
2 2
L
e− 4σ2 −σ Ω σ erfi 2σ
√
X0 = −
,
4L π
√
2 2
e−σ Ω − πσΩ erfc(σΩ)
A0 =
,
4π
(11.12)
(11.13)
where σ is the width of the window function (the time for which the detector is turned on),
and the subscripts indicate that these are the Minkowski vacuum results, with erfi(z) =
−i erf(iz) and erfc(z) = 1 − erf(z), where erf(z) is the error function. In the Minkowski
vacuum case, the detectors become entangled if and only if |X0 | > A0 . This region in the
L-Ω plane is above the slanted black line in Fig. 11.1.
Let’s see what happens with a nonzero temperature. Since we are interested in the
possibility that the perceived thermality is due to de Sitter expansion, we have a restriction
on the temperature, which sets the scale for the cosmic horizon LH = κ−1 = (2πT )−1 . If
observers are to exist at all, this horizon must be much larger than their typical scale of
experience, which can’t be much smaller than σ if the detector is to be useful to them.
(Consider how useful a “detector” that operates on the scale of the Hubble time would
be for humans.) Thus, for de Sitter expansion even to be a possibility, we require that
T σ −1 .
In both cases, the detector response function is given by Eq. (11.8), while the Wightman
function to be used in X in the thermal case is [185]
o
T n
+
Dth
xa (t); xb (t0 ) =
coth πT (L − y) + coth πT (L + y)
8πL
143
(11.14)
and in the de Sitter case is [154]
+
DdS
0
xa (t); xb (t ) =
−1
4π 2
sinh2 (πT y)
− e2πT x L2
π2T 2
−1
,
(11.15)
where x = t + t0 , and y = t − t0 − i in both. One can verify that in both cases, taking
L → 0 gives Eq. (11.8), and taking T → 0 gives Eq. (11.11).
In both the thermal and de Sitter cases, the integral in Eq. (11.10) can be well approxi+
mated by an asymptotic series in T (as T → 0), generated from the Taylor expansion of Dth
+
and DdS
, respectively, about x = y = 0. Although the radius of convergence of the Taylor
series is finite, for any reasonable detector setup, we are requiring that L σ. Since the
nearest pole is either O(L) or O(T −1 ) away, the Gaussian window function, whose width
is much smaller than either L or T −1 , will regularize, within the integral, any reasonably
truncated Taylor approximation to the Wightman function. This results in a valid asymptotic series for X in either case, as T → 0. The integral in Eq. (11.4) can be done similarly
by writing DT+ = D0+ + ∆DT+ (noting that the pole at y = 0 has been eliminated in ∆DT+ )
and calculating the temperature-dependent correction to Eq. (11.13). Numerical checks of
particular cases verify that these approximations are valid. The results are presented in
Fig. 11.1.
11.4
Discussion and Conclusion
Several points are in order here. First, detectors see anything at all in the Minkowski
vacuum case because the time-energy uncertainty relation, ∆t∆E &
1
2,
implies that a
detector operating for a finite time has a nonzero probability A0 of becoming excited, even
when the field is in the vacuum state. Entanglement exists when virtual particle exchange
dominates over local noise. When the magnitude of the exchange amplitude |X0 | exceeds
A0 , the detectors become entangled [170, 183]. Because of how both functions scale with Ω
and L, in the vacuum case one can always reduce the local noise below |X0 | by sufficiently
increasing Ω. In the thermal and de Sitter cases, the local noise profile A fails to decrease
fast enough for large Ω, resulting in a maximum entangling frequency for a given L, as well
as a maximum separation beyond which entanglement is impossible, regardless of Ω.
What does this mean for our curious planetary inhabitants? Let’s assume they have
two satellites, with detectors of the sort we’ve been using, located on comoving trajectories
as described above, with κ−1 < L < 2κ−1 so that in the de Sitter case they would be
outside of each other’s cosmic horizon but within that of the home planet (so they can
still send messages to it). The satellites are programmed to interact the field locally with
qubits having a resonant frequency that will lead to entanglement in the thermal case
and to a separable state in the de Sitter case (e.g., the red star in Fig. 11.1). After the
interactions, they each run a local measurement protocol that implements one side of a
144
entangled
only if T=0
3000
2500
entangled in
thermal case
but not de Sitter
ΣW
2000
entangled
in all 3 cases
1500
1000
ø
not
entangled
500
HhorizonL
0
0
500
1000
1500
2000
2500
3000
LΣ
Figure 11.1: Entanglement profile for detector pairs in several universes. σ is detection
time, Ω is detector resonance frequency, L is detector separation. The slanted black line is
the entanglement cutoff in the Minkowski vacuum case (entangled above, separable below).
The solid red curve is the thermal Minkowski cutoff, and the dashed blue curve is the de Sitter
vacuum cutoff, both with perceived local temperatures satisfying 2πT = 10−3 σ −1 . The de Sitter
horizon distance (103 σ) is given by the dotted green line. The red star indicates one particular
detector setup that could be used to distinguish expansion from heating.
145
test of Bell inequality violation, after which they send data back to the home planet for
analysis. If thermality is a result of expansion, there will be no entanglement, but if it is a
result of heating in flat spacetime, then the entanglement can be verified upon receipt of the
transmissions from both satellites. Because this effect only manifests when the detectors
pass beyond each others’ cosmic horizons (in the de Sitter case), a third party is required
to make the determination.
We have demonstrated that while expansion and heating give rise to the same (thermal)
signature in a single inertial particle detector, for certain choices of detector parameters, a
heated field in flat spacetime is able to entangle detector pairs that the conformal vacuum
in the associated de Sitter universe cannot. Thus, the universes can be distinguished by
their entangling power. Two detectors are required and must be beyond each others’ cosmic
horizons (in the de Sitter case) to see the effect. Although, if present, the entanglement is
exceedingly small, in principle its presence can always be determined by classical communication of local measurement data to a third party, as long as the verifier is able to receive
messages from both detectors. These results are contrary to the intuition that “curvature
generates entanglement” between field modes [176], since from it one would expect a larger
entangled region in the de Sitter case. The ability of the field to swap its entanglement to local detectors is an operational question, though, and for this setup, the vacuum in a curved
spacetime has less entangling power than a corresponding heated field in flat spacetime,
even though both produce the same local detector response.
146
Conclusion
147
Chapter 12
Why Quantum Information
Theory?
This thesis has explored many diverse topics: from quantum computation in its various
forms (Chapter 2) and, in depth, at the use of continuous-variable cluster states (Chapter 3),
including their compact optical implementation (Chapters 4, 5, and 6); through the use
of ion traps (Chapter 7) to simulate other quantum systems (Chapter 8) and precisely
study correlations within quantum states (Chapter 9); to an exploration of the nonintuitive
and nonclassical aspects of quantum theory, such as entanglement (Chapter 10) and its
application in the setting of curved-space quantum field theory (Chapter 11). But it began
with a story about a futuristic quantum computing platform (Chapter 1), affectionately
referred to as the “quantum cluster-computing core” or QCCC. The QCCC was envisioned
as the natural end product of the development of optical continuous-variable cluster-state
technology, as described in Part I.
Naturally, this narrative is far from a blueprint for a viable quantum computer, but it
was meant to portray certain key elements of the fictional technology. The obvious one is
that laser light was being used to quantum compute. Detectors were “clicking” possibly in
response to the adaptation of measurements based on previous results (see Chapter 3)—
changes that were being made with the aid of a classical computer (hence, the cables running
back and forth), a computer that need not be particularly powerful on its own but that
should be fast enough to handle high-speed feedback and control for a light-based system.
It is also conceivable that the QCCC may suffer from failures due to finite-squeezing errors
(Sec. 3.4) and that an adaptive scheme was being used, similar to that described in Sec. 3.5,
to discard a poor-performing state and try again with a new one. In this case, the clicking
might signify the repetitive preparation of new cluster states from the optical system, in
the vein of Chapter 6. Since it was being used to break (then-obsolete) RSA encryption, it
is almost certainly Shor’s factoring algorithm [15, 20, 25] that was being implemented by
the quantum computer.
148
Also emphasized in Chapter 1, however, is that fact that quantum information theory
is far richer than quantum computing alone. By viewing quantum mechanics as the processing of quantum information, questions that used to be about “stuff” can be recast in
information theoretic terms. Entanglement is a prime candidate for a concept that bridges
the gap between physical and information-theoretic concepts. In addition to its fundamental intrigue as a nonclassical phenomenon [11, 12, 33], entanglement is quantifiable entirely
in information theoretic terms [186, 187]. But entanglement is also a physical resource that
can be consumed for quantum teleportation [40] and one-way quantum computation using
cluster states [38] (also, Chapter 3). The one-way paradigm effectively replaces the physical
resource of laboratory equipment that could be used to perform coherent unitary evolution
with the physical resource of entanglement in the form of a cluster state (see Chapter 2).
This transformation naturally leads to questions of what the essential features are for
scalable quantum computation. It is known from the Gottesman-Knill Theorem [22, 36] and
its generalization to higher dimensions [51, 53] that entanglement alone is not enough, since
the Clifford group (of any dimension) can generate large-scale multipartite entanglement,
but quantum computations using only those operations are efficiently simulable on classical
computers and thus not universal (see Chapter 2). While entanglement is known to play
an important role [34, 188], other authors believe that Hilbert-space dimension [35]—apart
from any tensor-product separation (needed for a definition of entanglement [189])—is the
key. One can also come at the problem from the other direction: supposing that reality
allowed for efficient solution of certain classes of supposedly “difficult” problems, what
kind of a world would that be [190]? While these question remain open, this viewpoint
makes connections between information theory, computation, and physics that are rich and
inspiring.
With the quantum world seen as a playground for information theoretic concepts using
“quantum toys,” and with the interest in controlling quantum phenomena precisely for
applications to quantum computers, activities like nonclassical state preparation [101, 109]
and precise quantum control [110, 111] begin to take on a life of their own. Ion trap
technology [101, 102, 104] provides an ideal “quantum toy”—a simple quantum system able
to be controlled meticulously, with precise methods of detection (Part II). That term is
not meant in the slightest derogatory manner but is intended, rather, to elicit a sense of
discovery for its own sake—i.e., “Let’s see what we can do with this!” It turns out a lot
can be done, from quantum computation [106] to quantum simulations of all sorts of other
systems (Chapters 8 and 9).
Quantum information theory has answered a calling to provide a new perspective on
quantum foundations. It has been revealed in the last two decades that quantum theory
seems to be intimately connected to the physics of information [151, 153, 191], placing limits on what we can know and (in the case of entanglement) forcing us to abandon notions
of a “real state of affairs out there”—leaving us without the comforting belief that our
149
measurements are revealing preexisting properties of a system (see Section 10.1). While
many die-hard realists may prefer the de Broglie-Bohm [16, 17, 18] or other realistic (and
thus, nonlocal [11]) interpretation, the flavor of realist approaches are rooted in the idea of
the primacy of “stuff,” while other interpretations—the Bayesian interpretation [153] being
the primary example—mesh much more seamlessly within an information theoretic framework. The topic of interpretations of quantum theory frequently elicits opinions that share
more in common with religious zeal than with science, and this commentary is not meant
to definitely close the door on any of these views—including realist ones. It is intended,
however, to point out that there is a lesson to be learned from quantum information theory
and that some interpretations embrace this lesson, while others reject it. I believe that if
we are to make progress on the big questions in physics facing the community today, we
must embrace this lesson (see below).
Bayesians treat quantum states as serving the same function as probability distributions:
they quantify an agent’s willingness to place bets on outcomes of some future event [153]. In
fact, probability distributions are betting odds—one and the same. In the Bayesian view,
probabilities are not ontological—they do not exist in the world. Instead, they represent the
opinions of an agent and are rationally updated (via Bayes’ rule) but always contain an element of subjectivity. In the case of quantum theory, the “events” are future measurements,
the outcomes of which can be “bet” on. The way an agent assigns odds to those outcomes
constitute the quantum state she associates with the system to be measured. In this view,
the measurement “problem” is no such thing! Similarly, classical probability distributions
“collapse” after a measurement due to a logical update made by the agent assigning the
betting odds, based on new information obtained through the measurement. There is nothing dynamical about this procedure—it is a logical update, not a physical evolution of the
system in question. The only conceptual difference between the two—and the place where
die-hard realists choke—is that in the classical case, there remains the comforting ability
to believe that there is a “real state of affairs” of which the agent is ignorant; in quantum
theory—if we also wish to take relativity seriously with its principle of locality—this is not
allowed (see Section 10.1). The saving grace is that the idea of an agent’s ignorance is
independent of whether there is “something to be ignorant of.” The Bayesian mantra [153]:
Quantum theory is incomplete and cannot be completed.
This is not solipsism—there is a real world out there, to be sure. But there are aspects of
that world that are unknown and cannot be known—even in principle. What we can know
should be made consistent, and principles learned from other theories (e.g., relativity) should
be taken seriously. Physics, then, ceases to become the modeling of the movement of “stuff”
and instead becomes the structured understanding of what we can know about the world.
This is intimately related to the processes we use to obtaining information and also includes
communication, computation, and other abstract notions. Relations themselves become
150
more important than the “things” being related. This is the essence of an information
theoretic view of quantum theory.
This brings us to the most important contribution quantum information theory has
made to physics: a new interface to relativity theory. The relationship between quantum
theory and relativity is complex, to say the least, and entire research groups across the world
are actively devoted to work in this area. Most of the current effort focuses on string theory
(a.k.a. M-theory), but other efforts include loop quantum gravity, topos theory, twistor
theory, and others [192]. Somewhat surprisingly, when we specialize to the relationship
between quantum information theory and relativity, the landscape thins to a mere spotted
desert, boasting vast spaces of open problems broken up only by isolated results [19].
As research scientists, we are bound by rigor and objective verifiability of the claims we
make. On the other hand, when deciding on which problems would be worth examining,
one’s “gut” is as important as rationality. My gut tells me that this interface of quantum
information theory and relativity is where the solution to the problem of quantum gravity
will be found. I’m sure the hubris of this claim will offend some members of the other
communities, who have worked hard on this problem for many years. This is why I said,
“My gut tells me. . . .” It’s just a hunch. Nevertheless, it is this hunch that has led to my
interest in relativistic quantum information theory and the motivation for the toy project
described in Chapter 11.
Despite the speculative and presumptuous nature of the claim, there are reasons to believe it may be valid. First is the obvious fact that it is largely uncharted territory [19],
which on its own means little but still leads one to wonder what gems of discovery may lie
within it. Second is the intimate connection shown between quantum theory and information theory that has been revealed over the last two decades (discussed above). Third is
that information theoretic concepts are at the heart of some of the most interesting semiclassical results that we do have for phenomena at the interface of quantum theory and
relativity—e.g., Hawking radiation [160] and its relation to the holographic principle [172].1
Similar principles apply to Unruh-Davies radiation, particles detected by an accelerating
observer solely as a result of the acceleration itself [155, 156], and to Gibbons-Hawking
radiation, particles created from an expanding universe [123]. Finally, there is the historical
precedent that quite often a paradigm shift is required to overcome large problems that
persist through decades (or longer) of effort within the current mindset [153, 191, 195].
*
*
*
I truly hope you enjoyed reading this thesis, and I hope I was able to teach you some things
about quantum information theory and awaken you to the richness of the discipline. The
revolution from quantum theory as a physical theory to quantum theory as a theory of
information is still in its early stages, but I believe that this viewpoint holds the key to a
1
The holographic principle is often (incorrectly) touted as belonging only to the domain of string theory
in the form of the AdS/CFT correspondence [193, 194], which is only a particular instance of it.
151
needed paradigm shift. Regardless of whether you are persuaded by these claims to believe,
as I do, that quantum information theory holds the key to a revolution in physics, there
is no doubt that the landscape of quantum theory has been forever expanded. It’s time to
explore.
152
Appendix
153
Appendix A
Quantum Optics Cheat Sheet
This appendix provides a very brief summary of some of the basic concepts in quantum
optics. It is called a “cheat sheet” because it is not a proper introduction to the topics
at all, nor are all important topics included. It is intended only to be a short reminder of
what these topics are in the context of the work I present in the preceding chapters. For a
true introduction to quantum optics, the reader may wish to consult any quantum optics
textbook—for instance Ref. [54], which was used as my main reference. In addition, Ref. [99]
provides good coverage of topics in quantum optics specific to quantum information theory
using continuous variables.
A.1
A.1.1
Theory
Quantization of the electromagnetic field
The electromagnetic field is a quantized massless vector field with Lagrangian density
1
L(x) = − F µν Fµν ,
4
(A.1)
where F µν = ∂ µ Aν −∂ ν Aµ , with Aµ being the vector potential. Gauge freedom requires that
care be taken to quantize only the dynamical quantities. In the Coulomb gauge (∇ · A = 0)
and with box normalization, the quantized vector potential solution may be written as a
sum of plane-wave modes:
s
A(r, t) =
X
k,λ
i
~ h
ak,λ uk,λ (r)e−iωk t + a†k,λ u∗k,λ (r)eiωk t ,
2ωk 0
154
(A.2)
where k is the wave vector, λ indicates one of two orthogonal polarization states, and the
spatial mode functions uk,λ (r) are plane waves given by
uk,λ (r) =
eik·r
êλ ,
L3/2
(A.3)
where êλ is a (possibly complex) polarization unit-vector. Other spatial mode expansions
are possible. In all cases, however, the mode creation and annihilation operators satisfy the
canonical commutation relations for bosons,
[ak,λ , ak0 ,λ0 ] = δkk0 δλλ0 ,
(A.4)
and thus the modes behave as independent harmonic oscillators. As such, for each mode we
can define a “position” (also known as “quadrature-amplitude”) and “momentum” (a.k.a.
“quadrature-phase”) operator,
s
qk,λ =
r
and
pk,λ = −i
~
(ak,λ + a†k,λ )
2ωk
(A.5)
~ωk
(ak,λ − a†k,λ ) ,
2
(A.6)
respectively. These define a phase plane and allow use of standard tools for continuous
variables, such as the Wigner function (discussed next), to describe the states and dynamics
of the modes.
A.1.2
Wigner functions
The Wigner function is the quantum generalization of classical phase space. In both cases,
a function W (q, p) defines a state of the system. The main difference is that in the classical
case, the state is a probability distribution over the “actual state of affairs” in terms of a
microstate—the “real value” of position and momentum for each system; in the quantum
case, the Wigner function can take on negative values and thus cannot, in general, be
interpreted as a probability distribution.
One way to define the Wigner function for any state ρ is
1
W (q, p) :=
π~
Z
∞
ds hq − s| ρ |q + si e2ips/~ .
(A.7)
−∞
As for a classical phase-space distribution, the marginals of the Wigner function give probability distributions for one of the canonical variables (generalizations are also possible for
155
arbitrary measurements):
∞
Z
hq| ρ |qi = P (q) =
dp W (q, p) ,
(A.8)
dq W (q, p) .
(A.9)
Z−∞
∞
hp| ρ |pi = P (p) =
−∞
Fock states |ni, states of definite photon number, for n > 0 all have Wigner functions that
go negative; in contrast, all Gaussian states (discussed next) have positive Wigner functions.
A.1.3
Gaussian states, Gaussian operations, and the Heisenberg picture
A Gaussian state is any state whose Wigner function is a Gaussian distribution. Being
Gaussian, these states are uniquely specified by their mean and covariance matrix in the
phase plane. We can represent Gaussian states by an “error ellipse” corresponding to the
covariance matrix, centered at the mean. Several examples are illustrated in Figure A.1.
Gaussian operations are those that preserve the Gaussian character of a state, mapping
every Gaussian state to another Gaussian state.
In the Schrödinger picture, states evolve, and time-independent operators are constant.
In the Heisenberg picture, states are constant, and all time-evolution is represented in
the operators. If a quantum optical system only ever accesses Gaussian states, then its
evolution can be modeled easily in the latter. In this picture, the quadrature operators q
and p are treated as stochastic variables with fixed variance (usual equal to one unit of
vacuum noise). A Gaussian transformation is represented as a linear transformation on the
vector of quadrature operators (L is a matrix):
q0
!
p0
=L
q
!
p
+
∆q
!
∆p
.
(A.10)
Some simple operations on a single mode include displacement,
q 0 = q + ∆q ,
p0 = p + ∆p ,
(A.11)
phase-shift (rotation),
q0 =
q cos θ + p sin θ ,
p0 = −q sin θ + p cos θ ,
(A.12)
and squeezing,
q 0 = e+r q ,
p0 = e−r p .
156
(A.13)
p
(e)
(b)
(a)
(c)
q
(d)
Figure A.1: Examples of Gaussian states. The error ellipses for several single-mode Gaussian
states are drawn: (a) vacuum state, (b) thermal state (T > 0), (c) momentum-squeezed vacuum
state, (d) coherent state, (e) squeezed coherent state. The action of the harmonic Hamiltonian
on the states (for a single mode) is represented as counterclockwise rotation of the phase diagram
about the origin.
157
A common two-mode operation is interference through a beam splitter:
q10 =
q1 cos θ − p2 sin θ ,
q20 =
q2 cos θ − p1 sin θ ,
p01 =
q2 sin θ + p1 cos θ ,
p02 =
q1 sin θ + p2 cos θ ,
(A.14)
where sin θ = η is the amplitude reflectivity.
A.1.4
Single- and multi-mode-squeezed states
A “squeezed state” is any Gaussian state whose error ellipse is not spherical. Thus, there
exist linear combinations of the quadrature variables (“joint quadratures”) that have low
variance and other joint quadratures with high variance. When these low-variance—or,
“squeezed”—quadratures involve just a single mode, the state is said to be a single-modesqueezed state. The states labeled (c) and (e) in Figure A.1 are of this form, since the
plotted phase plane includes only one mode. When the squeezed quadratures involve more
than one mode (e.g., (p1 − q2 ) has a lowered variance), then the state is said to be a multimode-squeezed state. The latter type include states approximating those considered in the
EPR paradox [33], which would be “infinitely squeezed” since the variance of one joint
quadrature is made to vanish. In this optical picture, where q and p are quadratures of the
light field and not literal position and momentum, the ideal states considered by EPR have
infinite energy and are thus unphysical.
A.2
A.2.1
Experimental Implementation
Homodyne detection
Projective measurements on a quadrature variable (q cos θ + p sin θ) are very well approximated by the process of homodyne detection. A bright local oscillator beam, in a highintensity coherent state and phase-locked to the signal, is required. The two beams are
combined on a 50:50 beam splitter, and the outputs shone onto photodetectors (see Figure A.2). The resulting difference in the photocurrents is proportional to a measurement of
the quadrature variable written above, with θ adjustable by applying a phase-shift to the
local oscillator before the beam splitter. The constant of proportionality can be determined
by blocking the signal beam, thereby replacing it with the vacuum, whose variance is fixed.
To ensure proper phase-locking with the signal to be measured, it is important that the
local oscillator be split off from the main laser used by the experiment that generates the
signal.
158
signal
local
oscillator

phase
shift
Figure A.2: Schematic of homodyne detection. Homodyne detection involves the coherent
combination of the signal beam with a bright local oscillator on a 50:50 beam splitter, followed
by photodetection. The two beams must be phase-locked, which is normally accomplished
by generating both from the same laser. The difference in detected photocurrents closely
approximates projective measurement of a quadrature variable whose phase can be chosen by
phase-shifting the local oscillator before combination.
A.2.2
Squeezing—nonlinear media
Squeezing interactions can be accomplished using a nonlinear medium, which mediates a
Hamiltonian of the form
H ∝ i(ap a†s a†i − a†p as ai ) ,
(A.15)
where ap represents the pump beam, and as and ai , which are (somewhat arbitrarily) called
the signal and idler, correspond to the output modes. The phase θp of the pump beam
determines the quadrature being squeezed. Treating ap as a classical field, and absorbing its
intensity and the nonlinear susceptibility of the medium into an overall coupling constant κ,
gives an effective Hamiltonian of
H = i~κ(e−iθp a†s a†i − eiθp as ai ) .
(A.16)
The nonlinear medium interconverts pump photons and correlated signal-idler photon pairs,
generating squeezing between the latter. Energy conservation gives rise to the phasematching condition ωp = ωs + ωi , required for the interaction to take place. When the
signal and idler represent the same mode, single-mode (or “degenerate”) squeezing is accomplished; when the two represent different modes, two-mode (or “nondegenerate”) squeezing
159
� �����
� ����
�
� ����
M1
nonlinear crystal
� �����
M2
Figure A.3: Schematic of an optical parametric oscillator (OPO). An OPO consists of a
nonlinear medium placed within an optical cavity. The cavity mirrors M1 and M2 define
resonant modes, equally spaced in frequency, and indicated by the colored wavy lines. The
mirrors are partially reflecting and often chosen to be dichroic (reflecting some frequencies, while
transmitting others) at different frequencies. This allows input of the higher pump frequencies
from one side (green lines on the left), while outputting squeezed light at lower frequencies out
the other side (red lines on the right). The incoming, outgoing, and cavity modes are labeled
as shown.
takes place. In a typical nonlinear medium in free space, the interaction generates many
pairs of squeezed modes in a continuum of frequencies around ωp /2, while if it is placed
inside an optical cavity (discussed next), several interactions are still effected, but they are
discrete.
A.2.3
Optical parametric oscillator (OPO)
An OPO consists of a nonlinear medium in an optical cavity, as in Figure A.3. The nonlinear medium implements squeezing interactions as discussed above, while the cavity serves
to restrict the available modes to only the resonant ones, which are equally spaced in frequency, in analogy with the resonant modes of a taut string. A single pump frequency at
cavity mode ωp (where p is now an integer index) will generate a multitude of squeezing
interactions:
H = i~κ
X
(e−iθp a†j a†p−j − eiθp aj ap−j ) ,
(A.17)
j
where the starting index for j is chosen such that the phasematching condition, ωp =
ωj + ωp−j , is satisfied; the ending index is determined by the number of cavity modes within
the phasematching bandwidth of the interaction facilitated by the nonlinear medium.
The Hamiltonian in Eq. (A.17) does not completely describe the dynamics of the cavity
modes because it neglects the interaction of the internal modes with those outside the cavity
(by way of the partially reflecting mirrors). A full treatment of the intracavity dynamics
would include damping terms due to leakage out the endcaps. For linear Hamiltonians like
160
Eq. (A.17), the full equations of motion may be written as
1
d
1/2
1/2
a(t) = La(t) − (Γ1 + Γ2 )a(t) + Γ1 aL,in (t) + Γ2 aR,in (t) ,
dt
2
(A.18)
with boundary conditions
1/2
(A.19)
1/2
(A.20)
aL,in (t) + aL,out (t) = Γ1 a(t) ,
aR,in (t) + aR,out (t) = Γ2 a(t) ,
where a(t) and similar terms are column-vectors of creation and annihilation operators for
the modes as indicated in Figure A.3, L is the linear transformation effected by the interaction Hamiltonian, and Γ1,2 are diagonal matrices of damping coefficients for the associated
modes at mirrors 1 and 2, respectively. Solutions to these equations of motion often show a
threshold input power, above which the output beams are bright (stimulated emission) and
below which they are dark (spontaneous emission). When operating below threshold, the
dynamics are largely characterized by L—the Hamiltonian part of the interaction—although
the details, of course, depend on the particular experimental setup.
161
Mokey (1992–2006)
162
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