Download MULTIPARTICLE ENTANGLEMENT Sebastian Hartweg, Mario Berta (QSIT Seminar, 10.12.2012)

MULTIPARTICLE ENTANGLEMENT
Sebastian Hartweg, Mario Berta (QSIT Seminar, 10.12.2012)
Literature
Vol 438|1 December 2005|doi:10.1038/nature04251
LETTERS
Creation of a six-atom ‘Schrödinger cat’ state
D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1,
J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1
Among the classes of highly entangled states of multiple quantum
systems, the so-called ‘Schrödinger cat’ states are particularly
useful. Cat states are equal superpositions of two maximally
different quantum states. They are a fundamental resource in
fault-tolerant quantum computing1–3 and quantum communication, where they can enable protocols such as open-destination
teleportation4 and secret sharing5. They play a role in fundamental
tests of quantum mechanics6 and enable improved signal-to-noise
ratios in interferometry7. Cat states are very sensitive to decoherence, and as a result their preparation is challenging and can serve
as a demonstration of good quantum control. Here we report the
creation of cat states of up to six atomic qubits. Each qubit’s state
space is defined by two hyperfine ground states of a beryllium ion;
the cat state corresponds to an entangled equal superposition of all
the atoms in one hyperfine state and all atoms in the other
hyperfine state. In our experiments, the cat states are prepared
in a three-step process, irrespective of the number of entangled
atoms. Together with entangled states of a different class created
in Innsbruck8, this work represents the current state-of-the-art for
large entangled states in any qubit system.
One promising candidate system for scalable universal quantum
information processing (QIP) consists of atomic ions that are
confined in electromagnetic traps and manipulated with laser
beams9. Most of the basic ingredients for QIP10 have been demonstrated separately in the last few years in this system. Furthermore,
some simple algorithms that could serve as primitives for larger scale
QIP, including quantum error correction, teleportation, and the
S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We
define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN.
In this notation, prototypical cat states of N qubits can be written
as:
1
ð1Þ
jN Catl ¼ pffiffi ðj "; Nl þ eiv j #; NlÞ
2
To generate such states we initially prepare the ions in state j # , Nl
and then apply the following unitary operation to transform the
initial state into jN Catl (ref. 7):
$ h
"
#%& h
p i
yp
p i'& h p i'
exp i J 2z
U N ¼ exp i J x exp i J z
exp i J x
ð2Þ
2
2
2
2
The operators in the left and right pairs of parentheses represent a
common rotation by angle p2 of all N qubits, written in terms
of the global angular momentum operators !composed
P ! of the
sum of the N individual spin-1/2 operators J ¼ N
j¼1 Sj (Dicke
operators). The operator in the middle pair of parentheses
represents a global entangling interaction that is diagonal in the
measurement basis spanned by all product states of N qubits, each
in either j " l or j # l, and can be implemented by generalizing the
phase-gate mechanism described in ref. 14 (see also below). If N is
odd, y ¼ 1; y ¼ 0 otherwise.
Because of experimental imperfections, we need a measure to
indicate how close the generated state jW Nl is to the ideal state
jN Catl. The simplest measure, called the fidelity, is the square
modulus of the overlap of these two states:
2
Literature
Vol 438|1 December 2005|doi:10.1038/nature04251
LETTERS
Creation of a six-atom ‘Schrödinger cat’ state
D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1,
J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1
PRL
Among the classes of highly entangled states of multiple quantum
S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We
systems, the so-called ‘Schrödinger cat’ states are particularly
define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN.
In this notation, prototypical cat states of N qubits can be written
useful. Cat states are equal superpositions of two maximally
week ending
as:
different quantum states. They are a fundamental resource in
HY
S I C AcommuniL R E V I E W L E TjNTCatl
E R¼Sp1ffiffi ðj "; Nl þ eiv j #; NlÞ
1 APRIL
2011
106,
130506 quantum
(2011) computing1–3 Pand
quantum
fault-tolerant
ð1Þ
2
cation, where they can enable protocols such as open-destination
teleportation4 and secret sharing5. They play a role in fundamental
To generate such states we initially prepare the ions in state j # , Nl
tests of quantum mechanics6 and enable improved signal-to-noise
and then apply the following unitary operation to transform the
and
Coherence
states are veryEntanglement:
sensitive to decoher- Creation
ratios in interferometry7. Cat14-Qubit
initial state into
jN Catl
(ref. 7):
$ h
"
#%& h
ence, and as a result their preparation is challenging and can serve
i
p
p 2 i'& h p i'
1 of good quantum control.
1 Here we report the 1
1 i yp J
exp i1 William
U
exp
exp iA.J x Coish,
ð2Þ 2,3
¼
exp
i
J
J
as
a
demonstration
N
x
z
Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael
Chwalla,
Daniel
Nigg,
2
2
2 z
2
creation of cat states of up to six atomic qubits.
Each qubit’s state 4
1
1,*
1,4
Blatt
Harlander,
Hänsel,
Markus
Hennrich,
space is definedMaximilian
by two hyperfine
ground statesWolfgang
of a beryllium
ion;
The
operators
in the left and and
right Rainer
pairs of parentheses
represent a
1 corresponds to an entangled equal superposition of all
p
the cat state
common
rotation
by
angle
of
all
N
qubits,
Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020
Innsbruck, written
Austriain terms
2
2 in one hyperfine state and all atoms in the other
the atoms
of Physics
the globaland
angular
momentum
operators
composed
Institute for Quantum Computing and Department of
Astronomy,
University
of!Waterloo,
P ! of the
hyperfine state. In our experiments, the cat states are prepared
sum of the N individual spin-1/2 operators J ¼ N
j¼1 Sj (Dicke
Waterloo,
ON,
N2L
3G1,
Canada
in a three-step process,
operators). The operator in the middle pair of parentheses
3 irrespective of the number of entangled
of ofPhysics,
McGill
University,
Montreal,
Quebec,
Canada
H3A 2T8
atoms. Together with Department
entangled states
a different
class created
represents
a global
entangling
interaction
that is diagonal in the
4 8
für represents
Quantenoptik
und state-of-the-art
Quanteninformation,
Österreichische
Akademie
Wissenschaften,
, this work
the current
for
in InnsbruckInstitut
measurement
basis spanned
by all der
product
states of N qubits, each
large entangled states in any qubit system.
in either
j " l or j # l,
and can be implemented by generalizing the
Otto-Hittmair-Platz 1, A-6020
Innsbruck,
Austria
One promising candidate system(Received
for scalable 30
universal
quantum
mechanism
described
September
2010;phase-gate
published
31 March
2011)in ref. 14 (see also below). If N is
information processing (QIP) consists of atomic ions that are
odd, y ¼ 1; y ¼ 0 otherwise.
confined inWe
electromagnetic
traps and
manipulated with laser
Because
experimental
we need a measure
report the creation
of Greenberger-Horne-Zeilinger
statesofwith
up to 14 imperfections,
qubits. By investigating
the to
beams9. Most of the basic ingredients for QIP10 have been demonindicate how close the generated state jW Nl is to the ideal state
coherence
of
up
to
8
ions
over
time,
we
observe
a
decay
proportional
to
the
square
of
the
number
of
qubits.
strated separately in the last few years in this system. Furthermore,
jN Catl. The simplest measure, called the fidelity, is the square
Thealgorithms
observedthat
decay
a theoretical
modelmodulus
whichofassumes
a of
system
affected
some simple
couldagrees
serve as with
primitives
for larger scale
the overlap
these two
states: by correlated,
2
QIP, including quantum error correction, teleportation, and the
Literature
Vol 438|1 December 2005|doi:10.1038/nature04251
LETTERS
Creation of a six-atom ‘Schrödinger cat’ state
D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1,
J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1
Nobel Prize in Physics 2012
PRL
Among the classes of highly entangled states of multiple quantum
S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We
systems, the so-called ‘Schrödinger cat’ states are particularly
define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN.
In this notation, prototypical cat states of N qubits can be written
useful. Cat states are equal superpositions of two maximally
week ending
as:
different quantum states. They are a fundamental resource in
HY
S I C AcommuniL R E V I E W L E TjNTCatl
E R¼Sp1ffiffi ðj "; Nl þ eiv j #; NlÞ
1 APRIL
2011
106,
130506 quantum
(2011) computing1–3 Pand
quantum
fault-tolerant
ð1Þ
2
cation, where they can enable protocols such as open-destination
teleportation4 and secret sharing5. They play a role in fundamental
To generate such states we initially prepare the ions in state j # , Nl
tests of quantum mechanics6 and enable improved signal-to-noise
and then apply the following unitary operation to transform the
and
Coherence
states are veryEntanglement:
sensitive to decoher- Creation
ratios in interferometry7. Cat14-Qubit
initial state into
jN Catl
(ref. 7):
$ h
"
#%& h
ence, and as a result their preparation is challenging and can serve
i
p
p 2 i'& h p i'
1 of good quantum control.
1 Here we report the 1
1 i yp J
exp i1 William
U
exp
exp iA.J x Coish,
ð2Þ 2,3
¼
exp
i
J
J
as
a
demonstration
N
x
z
Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael
Chwalla,
Daniel
Nigg,
2
2
2 z
2
creation of cat states of up to six atomic qubits.
Each qubit’s state 4
1
1,*
1,4
Blatt
Harlander,
Hänsel,
Markus
Hennrich,
space is definedMaximilian
by two hyperfine
ground statesWolfgang
of a beryllium
ion;
The
operators
in the left and and
right Rainer
pairs of parentheses
represent a
1 corresponds to an entangled equal superposition of all
p
the cat state
common
rotation
by
angle
of
all
N
qubits,
Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020
Innsbruck, written
Austriain terms
2
2 in one hyperfine state and all atoms in the other
the atoms
of Physics
the globaland
angular
momentum
operators
composed
Institute for Quantum Computing and Department of
Astronomy,
University
of!Waterloo,
P ! of the
hyperfine state. In our experiments, the cat states are prepared
sum of the N individual spin-1/2 operators J ¼ N
j¼1 Sj (Dicke
Waterloo,
ON,
N2L
3G1,
Canada
in a three-step process,
operators). The operator in the middle pair of parentheses
3 irrespective of the number of entangled
of ofPhysics,
McGill
University,
Montreal,
Quebec,
Canada
H3A 2T8
atoms. Together with Department
entangled states
a different
class created
represents
a global
entangling
interaction
that is diagonal in the
4 8
für represents
Quantenoptik
und state-of-the-art
Quanteninformation,
Österreichische
Akademie
Wissenschaften,
, this work
the current
for
in InnsbruckInstitut
measurement
basis spanned
by all der
product
states of N qubits, each
large entangled states in any qubit system.
in either
j " l or j # l,
and can be implemented by generalizing the
Otto-Hittmair-Platz 1, A-6020
Innsbruck,
Austria
One promising candidate system(Received
for scalable 30
universal
quantum
mechanism
described
September
2010;phase-gate
published
31 March
2011)in ref. 14 (see also below). If N is
information processing (QIP) consists of atomic ions that are
odd, y ¼ 1; y ¼ 0 otherwise.
confined inWe
electromagnetic
traps and
manipulated with laser
Because
experimental
we need a measure
report the creation
of Greenberger-Horne-Zeilinger
statesofwith
up to 14 imperfections,
qubits. By investigating
the to
beams9. Most of the basic ingredients for QIP10 have been demonindicate how close the generated state jW Nl is to the ideal state
coherence
of
up
to
8
ions
over
time,
we
observe
a
decay
proportional
to
the
square
of
the
number
of
qubits.
strated separately in the last few years in this system. Furthermore,
jN Catl. The simplest measure, called the fidelity, is the square
Thealgorithms
observedthat
decay
a theoretical
modelmodulus
whichofassumes
a of
system
affected
some simple
couldagrees
serve as with
primitives
for larger scale
the overlap
these two
states: by correlated,
2
QIP, including quantum error correction, teleportation, and the
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Qubits in Ion Traps (I)
A string of ions in a linear Paul trap forms a quantum register.
Qubits in Ion Traps (I)
A string of ions in a linear Paul trap forms a quantum register.
Laser pulses can manipulate individual ions, or all ions
collectively.
Qubits in Ion Traps (I)
A string of ions in a linear Paul trap forms a quantum register.
Laser pulses can manipulate individual ions, or all ions
collectively.
2-level-ions can be realized:
● By electronic ground- and excited state.
● By hyperfine states of an electronic ground state.
Qubits in Ion Traps (II)
Read-out can be performed
by fluorescence detection.
Qubits in Ion Traps (II)
Read-out can be performed
by fluorescence detection.
Ion-ion interactions are
induced by laser excitation of
harmonic oscillations of ions in
the trap potential.
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multipartite Entanglement
Superdecoherence
Conclusions
Multiparticle Entanglement - ‘Cat States’
Superpositions of maximally different quantum states:
| #ii , | "ii
N qubits
| #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN
Multiparticle Entanglement - ‘Cat States’
Superpositions of maximally different quantum states:
| #ii , | "ii
N qubits
| #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN
1
p
|N, Cati =
| ", N i + ei✓ | #, N i
2
Cat state
Multiparticle Entanglement - ‘Cat States’
Superpositions of maximally different quantum states:
| #ii , | "ii
N qubits
| #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN
1
p
|N, Cati =
| ", N i + ei✓ | #, N i
2
Cat state
Important for fault tolerant quantum computation, quantum
communication, quantum simulations, metrology and quantum
to classical transition.
Multiparticle Entanglement - ‘Cat States’
Superpositions of maximally different quantum states:
| #ii , | "ii
N qubits
| #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN
1
p
|N, Cati =
| ", N i + ei✓ | #, N i
2
Cat state
Important for fault tolerant quantum computation, quantum
communication, quantum simulations, metrology and quantum
to classical transition.
Difficult to create → therefore used as demonstration of good
quantum control.
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Creation of Cat States - Blatt’s Group [2]
Optical transition in 40Ca+:
|S1/2 (m =
1/2)i ⌘ | #i |D5/2 (m =
1/2)i ⌘ | "i
[2] Phys. Rev. Lett. 106, 130506 (2011)
Creation of Cat States - Blatt’s Group [2]
Optical transition in 40Ca+:
|S1/2 (m =
1/2)i ⌘ | #i |D5/2 (m =
1/2)i ⌘ | "i
2-photon transitions in bichromatic laser field:
[2] Phys. Rev. Lett. 106, 130506 (2011)
Creation of Cat States - Wineland’s Group [1]
(I)
9Be+ ions, with hyperfine-splitting:
|F = 2, mF =
1i ⌘ | #i |F = 1, mF =
[1] Nature 438, 639 (2005)
1i ⌘ | "i
Creation of Cat States - Wineland’s Group [1]
(I)
9Be+ ions, with hyperfine-splitting:
|F = 2, mF =
1i ⌘ | #i |F = 1, mF =
Create | #, N i and apply unitary operator:
UN =
J~ =
✓
N
X
1i ⌘ | "i

◆⇣
h ⇡ i
h ⇡ i⌘ ⇣
h ⇡ i⌘
⇠⇡
2
exp i Jx exp i Jz
exp i Jz
exp i Jx
2
2
2
2
~j
S
Dicke oprators
j=1
⇠ = 0/1
N even/odd
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(I)
9Be+ ions, with hyperfine-splitting:
|F = 2, mF =
1i ⌘ | #i |F = 1, mF =
Create | #, N i and apply unitary operator:
UN =
J~ =
✓
N
X
1i ⌘ | "i

◆⇣
h ⇡ i
h ⇡ i⌘ ⇣
h ⇡ i⌘
⇠⇡
2
exp i Jx exp i Jz
exp i Jz
exp i Jx
2
2
2
2
~j
S
Dicke oprators
j=1
⇠ = 0/1
N even/odd
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(II)
Implementation of phase gate:
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(II)
Implementation of phase gate:
● Illumination of the ions with two
laser beams with detuning
!COM +
.
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(II)
Implementation of phase gate:
● Illumination of the ions with two
laser beams with detuning
!COM +
.
● Adjustment of detuning and pulse
length lead to a state dependent
phase shift.
| "i| "i| i ! | "i| "i| i
| "i| #i| i ! ei⇡/2 | "i| #i| i
| #i| "i| i ! ei⇡/2 | #i| "i| i
| #i| #i| i ! | #i| #i| i
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(III)
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(III)
First ⇡/2 -pulse:
| ##i = | #i ⌦ | #i
1
1
! p (| #i + | "i) ⌦ p (| #i + | "i)
2
2
1
= (| #i| #i + | "i| "i + | "i| #i + | #i| "i)
2
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(III)
First ⇡/2 -pulse:
| ##i = | #i ⌦ | #i
1
1
! p (| #i + | "i) ⌦ p (| #i + | "i)
2
2
1
= (| #i| #i + | "i| "i + | "i| #i + | #i| "i)
2
After phase pulse:
1
(| #i| #i + | "i| "i) |0i
2
+ ei (t) (| #i| "i|↵(t)i
+ | "i| #i|
↵(t)i)
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(III)
First ⇡/2 -pulse:
| ##i = | #i ⌦ | #i
1
1
! p (| #i + | "i) ⌦ p (| #i + | "i)
2
2
1
= (| #i| #i + | "i| "i + | "i| #i + | #i| "i)
2
After phase pulse:
1
(| #i| #i + | "i| "i) |0i
2
+ ei (t) (| #i| "i|↵(t)i
+ | "i| #i|
↵(t)i)
State after ⇡ and ⇡/2 pulse:
1
1
[(| #i| #i + | "i| "i)|0i + ei (t) (| #i| "i | "i| "i)
2
2
1
+ | #i| #i | "i| #i)|↵(t)i + ei (t) (| "i| #i | "i| "i
2
+ | #i| #i | #i| "i)| ↵(t)i]
[1] Nature 438, 639 (2005)
Creation of Cat States - Wineland’s Group [1]
(III)
First ⇡/2 -pulse:
| ##i = | #i ⌦ | #i
1
1
! p (| #i + | "i) ⌦ p (| #i + | "i)
2
2
1
= (| #i| #i + | "i| "i + | "i| #i + | #i| "i)
2
After phase pulse:
1
(| #i| #i + | "i| "i) |0i
2
+ ei (t) (| #i| "i|↵(t)i
+ | "i| #i|
↵(t)i)
State after ⇡ and ⇡/2 pulse:
1
1
[(| #i| #i + | "i| "i)|0i + ei (t) (| #i| "i | "i| "i)
2
2
1
+ | #i| #i | "i| #i)|↵(t)i + ei (t) (| "i| #i | "i| "i
2
+ | #i| #i | #i| "i)| ↵(t)i]
An ideal ⇡ displacement pulse leaves only | ""i and | ##i components!
[1] Nature 438, 639 (2005)
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Witnessing Multiparticle Entanglement (I)
Quantum state tomography? Entanglement measures?
Witnessing Multiparticle Entanglement (I)
Quantum state tomography? Entanglement measures?
Genuine N-particle entanglement.
Witnessing Multiparticle Entanglement (I)
Quantum state tomography? Entanglement measures?
Genuine N-particle entanglement.
‘Local operations and classical
communication’ (LOCC) entanglement
classes: cat state class.
Wikipedia
Witnessing Multiparticle Entanglement (I)
Quantum state tomography? Entanglement measures?
Genuine N-particle entanglement.
‘Local operations and classical
communication’ (LOCC) entanglement
classes: cat state class.
Wikipedia
Entanglement witness operator: expectation value negative entanglement.
Witnessing Multiparticle Entanglement (I)
Quantum state tomography? Entanglement measures?
Genuine N-particle entanglement.
‘Local operations and classical
communication’ (LOCC) entanglement
classes: cat state class.
Wikipedia
Entanglement witness operator: expectation value negative entanglement.
Method used in [1,2].
[1] Nature 438, 639 (2005) [2] Phys. Rev. Lett. 106, 130506 (2011)
Witnessing Multiparticle Entanglement (II)
⇢N
State
Witnessing Multiparticle Entanglement (II)
⇢N
W =1
State
2|N, CatihN, Cat| Entanglement witness
Witnessing Multiparticle Entanglement (II)
⇢N
W =1
State
2|N, CatihN, Cat| Entanglement witness
hW i⇢N = 1
2FN,Cat
Negative?
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
Witnessing Multiparticle Entanglement (II)
⇢N
W =1
State
2|N, CatihN, Cat| Entanglement witness
hW i⇢N = 1
2FN,Cat
Negative?
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
?
Witnessing Multiparticle Entanglement (II)
⇢N
W =1
State
2|N, CatihN, Cat| Entanglement witness
hW i⇢N = 1
Negative?
2FN,Cat
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
⇣ ⇡
exp i
4
j=1
N
O
(j)
=
(j)
x
(j)
⌘
cos +
?
Collectively rotate ⇢N
(j)
y
sin
Vary phase
Witnessing Multiparticle Entanglement (II)
⇢N
W =1
State
2|N, CatihN, Cat| Entanglement witness
hW i⇢N = 1
Negative?
2FN,Cat
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
⇣ ⇡
exp i
4
j=1
N
O
(j)
=
(j)
x
(j)
⌘
Collectively rotate ⇢N
cos +
P( ) = Peven ( )
?
(j)
y
sin
Podd ( )
A (P( )) = ⇢0...0,1...1
N
Vary phase
Probability of finding even /
odd number of excitations
Amplitude of oscillations
Witnessing Multiparticle Entanglement (II)
PRL 106, 130506 (2011)
PHYSICAL REVIEW LETT
Number
of qubits
1
0
⇢N
hW i⇢N = 1
⇣ ⇡
exp i
4
j=1
(j)
=
(j)
x
3
0
-1
1
Negative?
2FN,Cat
4
0
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
N
O
-1
1
(j)
⌘
?
-1
1
0
5
-1
1
Parity
W =1
State
2|N, CatihN, Cat| Entanglement witness
2
Collectively rotate ⇢N
6
0
-1
1
8
0
cos +
(j)
y
sin
Vary phase
-1
1
10
0
P( ) = Peven ( )
Podd ( )
A (P( )) = ⇢0...0,1...1
N
Probability of finding even /
odd number of excitations
Amplitude of oscillations
-1
1
12
0
-1
1
0
-1
14
0
0.2
0.4
0.6
0.8
Phase φ of analyzing pulse (π)
1
1 (color
observed
on
[2]FIG.
Phys.
Rev.online).
Lett.Parity
106,oscillations
130506
(2011)
f2; 3; 4; 5; 6; 8; 10; 12; 14g-qubit GHZ states.
The coherence of GHZ states as a function of time is
FIG. 2 (color o
probability !ðNÞ
function of time
2 (green), 3 (red
observed relative
behavior proport
coherence of an
faster than the co
magnetic field d
ing coils. By de
ence time impro
coherence time
Witnessing Multiparticle Entanglement (II)
PRL 106, 130506 (2011)
PHYSICAL REVIEW LETT
Number
of qubits
1
0
⇢N
hW i⇢N = 1
⇣ ⇡
exp i
4
j=1
(j)
=
(j)
x
3
0
-1
1
Negative?
2FN,Cat
4
0
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
N
O
-1
1
(j)
⌘
?
-1
1
0
5
-1
1
Parity
W =1
State
2|N, CatihN, Cat| Entanglement witness
2
Collectively rotate ⇢N
6
0
-1
1
8
0
cos +
(j)
y
sin
Vary phase
-1
1
10
0
P( ) = Peven ( )
Podd ( )
A (P( )) = ⇢0...0,1...1
N
Probability of finding even /
odd number of excitations
Amplitude of oscillations
-1
1
-1
1
0
-1
Attention: rotations and measurements reliable?
12
0
14
0
0.2
0.4
0.6
0.8
Phase φ of analyzing pulse (π)
1
1 (color
observed
on
[2]FIG.
Phys.
Rev.online).
Lett.Parity
106,oscillations
130506
(2011)
f2; 3; 4; 5; 6; 8; 10; 12; 14g-qubit GHZ states.
The coherence of GHZ states as a function of time is
FIG. 2 (color o
probability !ðNÞ
function of time
2 (green), 3 (red
observed relative
behavior proport
coherence of an
faster than the co
magnetic field d
ing coils. By de
ence time impro
coherence time
all ions. The electronic and vibrational states of the ion
string are manipulated by setting the frequency, duration,
intensity, and phase of the pulses. Finally, the state of the
ion qubits is measured by scattering light at 397 nm on the
S1=2 $ P1=2 transition and detecting the fluorescence with
a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual
qubit in the fj0i; j1ig basis, while the PMT only detects the
number of ions being in j0i or j1i. Sufficient statistics is
achieved by repeating each experiment 100 times for each
setting.
f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed
parity oscillations are shown in Fig. 1. Although N-particle
distillability can be inferred from the criterion in Ref. [14]
by many standard deviations, according to the criteria in
Ref. [15] the obtained data support genuine N-particle
entanglement for 14 qubits with a confidence of 76%.
The 12-qubit state is likely not fully entangled. The
Poissonian statistics of the PMT fluorescence data is
accounted for by a data analysis based on Bayesian
inference [16].
Witnessing Multiparticle Entanglement (III)
Experimental results from [2]:
TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria
supported by & standard deviations. All errors in parenthesis, 1 standard deviation.
Number of ions
Populations, %
Coherence, %
Fidelity, %
Distillability criterion [14], &
Entanglement criterion [15], &
2
3
4
5
6
8
10
12
14
99.50(7)
97.8(3)
98.6(2)
283
265
97.6(2)
96.5(6)
97.0(3)
151
143
97.5(2)
93.9(5)
95.7(3)
181
167
96.0(4)
92.9(8)
94.4(5)
100
101
91.6(4)
86.8(8)
89.2(4)
95
96
84.7(4)
78.7(7)
81.7(4)
96
92
67.0(8)
58.2(9)
62.6(6)
40
25
53.3(9)
41.6(10)
47.4(7)
18
$6
56.2(11)
45.4(13)
50.8(9)
17
0.7
130506-2
[2] Phys. Rev. Lett. 106, 130506 (2011)
all ions. The electronic and vibrational states of the ion
string are manipulated by setting the frequency, duration,
intensity, and phase of the pulses. Finally, the state of the
ion qubits is measured by scattering light at 397 nm on the
S1=2 $ P1=2 transition and detecting the fluorescence with
a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual
qubit in the fj0i; j1ig basis, while the PMT only detects the
number of ions being in j0i or j1i. Sufficient statistics is
achieved by repeating each experiment 100 times for each
setting.
f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed
parity oscillations are shown in Fig. 1. Although N-particle
distillability can be inferred from the criterion in Ref. [14]
by many standard deviations, according to the criteria in
Ref. [15] the obtained data support genuine N-particle
entanglement for 14 qubits with a confidence of 76%.
The 12-qubit state is likely not fully entangled. The
Poissonian statistics of the PMT fluorescence data is
accounted for by a data analysis based on Bayesian
inference [16].
Witnessing Multiparticle Entanglement (III)
Experimental results from [2]:
TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria
supported by & standard deviations. All errors in parenthesis, 1 standard deviation.
Number of ions
Populations, %
Coherence, %
Fidelity, %
Distillability criterion [14], &
Entanglement criterion [15], &
2
3
4
5
6
8
10
12
14
99.50(7)
97.8(3)
98.6(2)
283
265
97.6(2)
96.5(6)
97.0(3)
151
143
97.5(2)
93.9(5)
95.7(3)
181
167
96.0(4)
92.9(8)
94.4(5)
100
101
91.6(4)
86.8(8)
89.2(4)
95
96
84.7(4)
78.7(7)
81.7(4)
96
92
67.0(8)
58.2(9)
62.6(6)
40
25
53.3(9)
41.6(10)
47.4(7)
18
$6
56.2(11)
45.4(13)
50.8(9)
17
0.7
130506-2
[2] Phys. Rev. Lett. 106, 130506 (2011)
all ions. The electronic and vibrational states of the ion
string are manipulated by setting the frequency, duration,
intensity, and phase of the pulses. Finally, the state of the
ion qubits is measured by scattering light at 397 nm on the
S1=2 $ P1=2 transition and detecting the fluorescence with
a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual
qubit in the fj0i; j1ig basis, while the PMT only detects the
number of ions being in j0i or j1i. Sufficient statistics is
achieved by repeating each experiment 100 times for each
setting.
f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed
parity oscillations are shown in Fig. 1. Although N-particle
distillability can be inferred from the criterion in Ref. [14]
by many standard deviations, according to the criteria in
Ref. [15] the obtained data support genuine N-particle
entanglement for 14 qubits with a confidence of 76%.
The 12-qubit state is likely not fully entangled. The
Poissonian statistics of the PMT fluorescence data is
accounted for by a data analysis based on Bayesian
inference [16].
Witnessing Multiparticle Entanglement (III)
Experimental results from [2]:
TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria
supported by & standard deviations. All errors in parenthesis, 1 standard deviation.
Number of ions
Populations, %
Coherence, %
Fidelity, %
Distillability criterion [14], &
Entanglement criterion [15], &
2
3
4
5
6
8
10
12
14
99.50(7)
97.8(3)
98.6(2)
283
265
97.6(2)
96.5(6)
97.0(3)
151
143
97.5(2)
93.9(5)
95.7(3)
181
167
96.0(4)
92.9(8)
94.4(5)
100
101
91.6(4)
86.8(8)
89.2(4)
95
96
84.7(4)
78.7(7)
81.7(4)
96
92
67.0(8)
58.2(9)
62.6(6)
40
25
53.3(9)
41.6(10)
47.4(7)
18
$6
56.2(11)
45.4(13)
50.8(9)
17
0.7
130506-2
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
hW i⇢N = 1
[2] Phys. Rev. Lett. 106, 130506 (2011)
2FN,Cat
all ions. The electronic and vibrational states of the ion
string are manipulated by setting the frequency, duration,
intensity, and phase of the pulses. Finally, the state of the
ion qubits is measured by scattering light at 397 nm on the
S1=2 $ P1=2 transition and detecting the fluorescence with
a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual
qubit in the fj0i; j1ig basis, while the PMT only detects the
number of ions being in j0i or j1i. Sufficient statistics is
achieved by repeating each experiment 100 times for each
setting.
f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed
parity oscillations are shown in Fig. 1. Although N-particle
distillability can be inferred from the criterion in Ref. [14]
by many standard deviations, according to the criteria in
Ref. [15] the obtained data support genuine N-particle
entanglement for 14 qubits with a confidence of 76%.
The 12-qubit state is likely not fully entangled. The
Poissonian statistics of the PMT fluorescence data is
accounted for by a data analysis based on Bayesian
inference [16].
Witnessing Multiparticle Entanglement (III)
Experimental results from [2]:
TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria
supported by & standard deviations. All errors in parenthesis, 1 standard deviation.
Number of ions
Populations, %
Coherence, %
Fidelity, %
Distillability criterion [14], &
Entanglement criterion [15], &
2
3
4
5
6
8
10
12
14
99.50(7)
97.8(3)
98.6(2)
283
265
97.6(2)
96.5(6)
97.0(3)
151
143
97.5(2)
93.9(5)
95.7(3)
181
167
96.0(4)
92.9(8)
94.4(5)
100
101
91.6(4)
86.8(8)
89.2(4)
95
96
84.7(4)
78.7(7)
81.7(4)
96
92
67.0(8)
58.2(9)
62.6(6)
40
25
53.3(9)
41.6(10)
47.4(7)
18
$6
56.2(11)
45.4(13)
50.8(9)
17
0.7
130506-2
FN,Cat = F (⇢N , |N, CatihN, Cat|)
⌘
1 ⇣ 0...0,0...0
1...1,1...1
=
⇢N
+ ⇢N
+ ⇢0...0,1...1
N
2
hW i⇢N = 1
Other methods.
[2] Phys. Rev. Lett. 106, 130506 (2011)
2FN,Cat
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Superdecoherence (I)
Decoherence (fidelity)?
PRL 106, 130506 (2011)
Superdecoherence (I)
PHYSICAL REVIEW LETTERS
week ending
1 APRIL 2011
Number
of qubits
1
0
-1
1
0
2
Decoherence (fidelity)?
3
-1
1
0
4
-1
1
0
5
Parity
-1
1
0
6
-1
1
0
8
-1
1
0
-1
1
0
10
[2] Phys. Rev. Lett. 106, 130506 (2011)
12
FIG. 2 (color online). Coherence decay and relative error
PRL 106, 130506 (2011)
Superdecoherence (I)
PHYSICAL REVIEW LETTERS
week ending
1 APRIL 2011
Number
of qubits
1
0
-1
1
0
-1
1
0
2
Decoherence (fidelity)?
Gate time: ⇠ 100µs.
3
4
-1
1
0
5
Parity
-1
1
0
6
-1
1
0
8
-1
1
0
-1
1
0
10
[2] Phys. Rev. Lett. 106, 130506 (2011)
12
FIG. 2 (color online). Coherence decay and relative error
PRL 106, 130506 (2011)
Superdecoherence (I)
PHYSICAL REVIEW LETTERS
week ending
1 APRIL 2011
Number
of qubits
1
0
-1
1
0
-1
1
0
-1
1
0
Parity
-1
1
0
2
Decoherence (fidelity)?
Gate time: ⇠ 100µs.
3
Sources of noise: magnetic
field noise, laser beam
fluctuations, spontaneous
emission, ...
4
5
6
-1
1
0
8
-1
1
0
-1
1
0
10
[2] Phys. Rev. Lett. 106, 130506 (2011)
12
FIG. 2 (color online). Coherence decay and relative error
PRL
PRL106,
106,130506
130506(2011)
(2011)
Superdecoherence (I)
PPHHYYSSI ICCAALL RREEVVIIEEW
W LLEETTTTEERRSS
week
week ending
11 APRIL
APRIL 2011
Number
Number
qubits
ofofqubits
11
00
-1 -1
11
00
-1 -1
11
00
-1 -1
11
00
Parity
Parity
-1 -1
11
00
-1 -1
11
00
22
Decoherence (fidelity)?
Gate time: ⇠ 100µs.
33
Sources of noise: magnetic
field noise, laser beam
fluctuations, spontaneous
emission, ...
Decay proportional N 2.0(1),
superdecoherence [2]!
44
55
66
88
-1 -1
11
00
-1 -1
1
1
0
0
10
10
[2] Phys. Rev. Lett. 106, 130506 (2011)
12
12
FIG.22 (color
(color online).
online). Coherence
Coherence decay
decay and
and relative
relative error
error
FIG.
PRL
PRL106,
106,130506
130506(2011)
(2011)
Superdecoherence (I)
PPHHYYSSI ICCAALL RREEVVIIEEW
W LLEETTTTEERRSS
week
week ending
11 APRIL
APRIL 2011
Number
Number
qubits
ofofqubits
11
00
-1 -1
11
00
-1 -1
11
00
-1 -1
11
00
Parity
Parity
-1 -1
11
00
-1 -1
11
00
-1 -1
11
00
-1 -1
1
1
0
0
22
Decoherence (fidelity)?
Gate time: ⇠ 100µs.
33
Sources of noise: magnetic
field noise, laser beam
fluctuations, spontaneous
emission, ...
Decay proportional N 2.0(1),
superdecoherence [2]!
44
55
66
Why not N?
88
10
10
[2] Phys. Rev. Lett. 106, 130506 (2011)
12
12
FIG.22 (color
(color online).
online). Coherence
Coherence decay
decay and
and relative
relative error
error
FIG.
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
[2] Phys. Rev. Lett. 106, 130506 (2011)
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
Magnetic field noise accounts for collective phase errors:
N
Hnoise =
E(t) X
2 j=1
(j)
z
[2] Phys. Rev. Lett. 106, 130506 (2011)
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
Magnetic field noise accounts for collective phase errors:
N
Hnoise =
E(t) X
2 j=1
(j)
z
For collective Gaussian phase noise it can be shown that:
F (N ) /
F (1)
N2
[2] Phys. Rev. Lett. 106, 130506 (2011)
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
Magnetic field noise accounts for collective phase errors:
N
Hnoise =
E(t) X
2 j=1
(j)
z
For collective Gaussian phase noise it can be shown that:
F (N ) /
F (1)
N2
Decoherence free subspaces:
[2] Phys. Rev. Lett. 106, 130506 (2011)
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
Magnetic field noise accounts for collective phase errors:
N
Hnoise =
E(t) X
2 j=1
(j)
z
For collective Gaussian phase noise it can be shown that:
F (N ) /
F (1)
N2
1
p (| ####""""i + | """"####i)
2
Decoherence free subspaces:
has a decoherence time of 324(42) ms [2]!
[2] Phys. Rev. Lett. 106, 130506 (2011)
Superdecoherence (II)
Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms
coherence time for single qubit [2]!
Magnetic field noise accounts for collective phase errors:
N
Hnoise =
E(t) X
2 j=1
(j)
z
Collective Gaussian phase noise
from magnetic field fluctuations
For collective Gaussian phase noise it can be shown that:
F (N ) /
F (1)
N2
1
p (| ####""""i + | """"####i)
2
Decoherence free subspaces:
has a decoherence time of 324(42) ms [2]!
[2] Phys. Rev. Lett. 106, 130506 (2011)
Outline
Qubits in Ion Traps
Multiparticle Entanglement - ‘Cat States’
Creation of ‘Cat States’
Witnessing Multiparticle Entanglement
Superdecoherence
Conclusions
Conclusions
Create cat states of up to 14 trapped ions.
Cat states are crucial for manifold applications in quantum
information science.
Collective phase noise, superdecoherence. Use decoherence
free subspaces.
A priori technical problem to reduce noise, but scalability
problematic. Create Schrödinger’s cat?
Check out:
http://www.nobelprize.org/nobel_prizes/physics/
PHYSICS:
Particle control in a quantum world
09:10–09:40 Controlling photons in a box and exploring
the quantum to classical boundary
SERGE HAROCHE, COLLÈGE DE FRANCE AND
ECOLE NORMALE SUPÉRIEURE, PARIS, FRANCE
09:45–10:15 Superposition, entanglement,
and raising Schroedinger’s cat
DAVID J. WINELAND, NATIONAL INSTITUTE OF
STANDARDS AND TECHNOLOGY (NIST) AND
UNIVERSITY OF COLORADO BOULDER, CO, USA
CHEMISTRY:
Check out:
http://www.nobelprize.org/nobel_prizes/physics/
PHYSICS:
Particle control in a quantum world
09:10–09:40 Controlling photons in a box and exploring
the quantum to classical boundary
SERGE HAROCHE, COLLÈGE DE FRANCE AND
ECOLE NORMALE SUPÉRIEURE, PARIS, FRANCE
09:45–10:15 Superposition, entanglement,
and raising Schroedinger’s cat
DAVID J. WINELAND, NATIONAL INSTITUTE OF
STANDARDS AND TECHNOLOGY (NIST) AND
UNIVERSITY OF COLORADO BOULDER, CO, USA
CHEMISTRY:
That’s it.