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Project submitted for the base funding of
Artem Alikhanyan National Laboratory (ANL)
Principal Investigator:
Sayatnova Tamaryan
TITLE:
Quantum Computation and Communication Technology
Division, group:
Theory Department
DURATION:
three years
Estimated Project Costs (± 20%)
Estimated total cost of the project (US $) 94 300
Including:
Payments to Individual Participants
75 600
Equipment
6 200
Materials
2 500
Other Direct Costs
2 000
Travel
8 000
PROBLEM:
Building quantum information processing devices is a great challenge for scientists and engineers
of the third millennium. Compound quantum systems have potential for many quantum processes,
including remarkable applications: exponential speedup of quantum computers, perfect teleportation,
superdense coding,
quantum cryptography and the factoring of large composite numbers. These
phenomena have provided a basis for the development of modern quantum information science. But
most of the surprises are still ahead of us. Quantum computation and communication, and more generally
quantum information science and technologies, are here to stay and will have a profound impact on the
XXI century.
The superior performance of quantum systems in computation and communication applications is
predominantly rooted in a property of quantum mechanical states called entanglement. Essentially,
entanglement comes along with new kinds of correlations. It is a fundamental property of quantum
systems and a basic physical resource for Quantum Information, Computation and Communication. It is
increasingly realized that quantum entanglement is at the heart of quantum physics and as such it may
be of very broad importance for modern science and future technologies. Hence quantifying the
entanglement of multipartite quantum states is a problem of vital importance.
1
OBJECTIVES:
Electromagnetic and strong-weak dualities provide deep insight into the nature of quantum field theories.
Here we discover a new analogue of these dualities, hereinafter referred to as inverse duality, which is an
inherent feature of the multipartite quantum entanglement. We use this duality to compute analytically the
geometric measure of entanglement of general multipartite quantum states. Additionally, new duality
reveals two critical values for the quantum entanglement and shows that pure quantum states at critical
values are capable to teleport arbitrary multipartite states including mesoscopic and nanoscale systems.
On the basis of this result we will suggest an experimentally accessible protocol for the perfect
teleportation of an unknown multi-qubit state. The protocol allows us to construct a long-distance
quantum communication channel of high capacity. Next we will extend the two basic operational
entanglement measures, namely the entanglement cost and the distillable entanglement, to three-qubit
states. We will computed analytically these measures, as well as the relative entropy of entanglement,
and use these tools to analyze the behavior of quantum states at the critical values of the quantum
entanglement.
TASK 1: Duality and critical values of quantum entanglement
Task description and main milestones
Participating Institutions
Task 1.1 Double maps between entangled and separable states
Task 1.2 Double maps between inequivalent classes of entanglement
Task 1.3 Stationary points of dual transformations
University of York, Britain
Description of deliverables
1. Inverse duality transformations for quantum entanglement,
2. Critical values of multipartite entanglement
TASK 2: Teleportation of multipartite states through noisy channels
Task description and main milestones
Participating Institutions
Task 2.1 Multipartite pure states applicable for perfect teleportation
Task 2.2 Measurement basis, local operations assisted by classical
communication
Task 2.3 Teleportation through noisy channels via mixed states
Kyungnam University,
Korea
Description of deliverables
1. Explicit protocol for teleportation of multipartite states through noisy channels
2. Long-distance quantum communication channel of high capacity
2
TASK 3: Tripartite quantum states at criticality
Task description and main milestones
Participating Institutions
Task 3.1 Distillation and dilution of the three-qubit entanglement
Task 3.2 Reduced stationarity equations for the relative entropy
of entanglement
University of British Columbia,
Canada
Description of deliverables
1. The entanglement cost and distillable entanglement of three-qubit states
2. Relative entropy of entanglement of pure three-qubit states
IMPACT:
Quantum information science is a new field of science and technology, combining and drawing
on the disciplines of physical science, mathematics, computer science, and engineering. Its aim is to
understand how certain fundamental laws of physics can be harnessed to dramatically improve the
acquisition, transmission, and processing of information. The exciting scientific opportunities offered by
quantum information science are attracting the interest of a growing community of scientists and
technologists, and are promoting unprecedented interactions across traditional disciplinary boundaries.
Advances in quantum information will become increasingly critical to our national competitiveness in
information technology during this century.
The teleportation schemes discovered so far were able to teleport one- or two-particle states,
while our scheme can teleport quantum states with unlimited number of particles. Owing to this we can
construct a quantum channel of high capacity for secure communications. Therefore, we think the project
is highly important for science and technology in Armenia.
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Brief survey of the worldwide researches made on the project topic, the
competitiveness of the project, and achievements of the group (not more than 2
pages):
Entanglement was first described by Einstein, Podolsky, and Rosen [1] and Schrodinger
[2] as a strange phenomenon of quantum mechanics, questioning the completeness of the
theory. Later, Bell recognized that entanglement leads to experimentally testable deviations of
quantum mechanics from classical physics [3]. Finally, with the advent of quantum information
theory, entanglement was recognized as a resource, enabling tasks like the factoring of large
composite numbers [4,5], exponential speed-up of quantum computers quantum [6,7,8],
quantum cryptography [9,10], quantum teleportation [11,12] and superdense coding [13,14].
Together with the rapid experimental progress on quantum control, this lead to a rapidly
growing interest in entanglement theory and many experiments nowadays aim at the
generation of entanglement.
The quantification of entanglement of multipartite pure states is the most important
problem in quantum information theory. Intensive studies are under way, and different
entanglement measures have been proposed over the years [15-19]. However, it is generally
impossible to calculate their value because the definition of any multipartite entanglement
measure usually includes a massive optimization over certain quantum protocols or states
[20,21,22].
Our team (together with Prof. Sudbery from the University of York) has made a
breakthrough in the understanding of quantum entanglement of multipartite systems. An
intimate connection between quantum entanglement [23,24] and duality [25-28] has emerged.
The essence of the new duality, referred to as inverse duality, is the following.
Three-qubit states can be entangled in two inequivalent ways and consist of
Greenberger-Horne-Zeilinger(GHZ)-class and W-class states [29]. These classes are mutually
exclusive and exhaustive and cannot be transformed to each other via local operations and
classical communication nor even stochastically. The inverse duality transformation maps a
given GHZ-class state to a W-class state and relates entanglements of those states from the
different classes. It has three important consequences.
First, if one knows the geometric measure of entanglement (GM) [17-19,22] of an
arbitrary GHZ- or W-class state, then one instantly finds GM of its image from the opposite
class. On the other hand GM of general W states has been computed analytically [30] and
therefore GM of general GHZ-class states can also be computed using the inverse duality. This
task we will carry out together with Prof. Sudbery. Notice, so far Ref.[30] is a unique paper,
where an entanglement measure is computed analytically for a class of general n-qubit states.
Second, a duality between pure W states and separable states has already been
constructed in Ref.[30] and it has revealed two critical points for GM of W states. Surprisingly,
4
GM of many-qubit W states is universal between two critical points since the first critical point
imposes a unique behavior on GM [31]. In fact GM depends only on the Bloch vector with the
minimal z-component. Hence one can prepare many-qubit W states, among them lowdimensional, mesoscpic and nanosclae W states, with the required entanglement by altering
the Bloch vector of a single qubit.
Third, at the second critical point the reduced density operator of one qubit is a constant
multiple of the unit operator and then GM of the W state is absolutely independent of the state
parameters. The existence of critical points was already proposed in Ref.[22] , proved for
general W states in Ref.[30] and extended to arbitrary three-qubit states in Ref.[32]. Pure
three-qubit states at this point, known as shared quantum states [33], can be used as a
quantum channel for the perfect teleportation (dense coding) [11,12,34-37]. Therefore using
critical points of multipartite entanglement one can generalize teleportation protocol to arbitrary
multi-qubit states. This task we will carry out together with Prof. Park from the Kyungnam
University.
On the basis of the multipartite teleportation we will elaborate a quantum communication
technology capable to transmit an unlimited amount of information through noisy channels.
Note, that a long-distance quantum communication channel of high capacity enables us to
solve several problems of practical importance.
Next task of this project is to extend the basic operational entanglement measures, i.e.
the entanglement cost and the distillable entanglement [23,24], to three-qubit states and
compute them analytically. The main motivation is that, the behavior of GM of mesoscopic and
nanoscale systems is universal between the two critical points [31] and it is unclear is it a
specific feature of GM or is it a general feature of an arbitrary entanglement measure? Hence
we (together with Prof. Wei from the University of British Columbia) will generalize famous
Wootters's concurrence [20] to tripartite systems and use it to fully understand the role of the
critical values of quantum entanglement. We have already made a major step toward the
generalization in Ref.[38], where the three-qubit maximally entangled states are derived and
classified by a gauge phase.
The progress made to date allows oneself to calculate analytically the entanglement for
pure three-qubit systems [39]. The basic idea is to use (n-1)- qubit mixed states to calculate the
entanglement of n-qubit pure states [40]. In the case of three qubits this idea converts
effectively the nonlinear eigenvalue problem into linear eigenvalue equations. Solutions derived
in this way give analytic expressions for GM [22]. Note, non of other methods were able to give
explicit solutions for generic three-qubit systems. The developed effective method can be
applied to other entanglement measures
and, among them, to the relative entropy of
entanglement. The research in this direction was already initiated by our Korean partners [41],
where the relative entropy of entanglement of low-rank states is computed.
5
References:
1. A. Einstein, B. Podolsky, and N. Rosen,
Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,
Phys. Rev. 47, 777 (1935).
2. E. Schrodinger, Die gegenwärtige Situation in der Quantenmechanik,
Die Naturwissenschaften, 23, 807 (1935).
3. J. S. Bell, On the Einstein-Podolsky-Rosen Paradox,
Physics 1, 195 (1964).
4. P. Shor, Algorithms for Quantum Computation: Discrete Logarithm and Factoring,
Proc. 35th Annual Symposium on Foundations of Computer Science, 124-134 (1994).
5. L. M.K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang,
Experimenta realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,
Nature 414, 883 (2001).
6. G. Vidal, Efficient Classical Simulation of Slightly Entangled Quantum Computations,
Phys. Rev. Lett. 91, 147902 (2003).
7. R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up,
Proc. R. Soc. Lond. A 459, 2011 (2003).
8. C. Negrevergne, T. S. Mahesh, C. A. Ryan, M. Ditty, F. Cyr-Racine, W. Power, N. Boulant, T. Havel,
D. G.Cory, and R. Laflamme, Benchmarking Quantum Control Methods on a 12-Qubit System,
Phys. Rev. Lett. 96, 170501 (2006).
9. K. Ekert, Quantum cryptography based on Bell’s theorem,
Phys. Rev. Lett 67, 661 (1991).
10. C. H. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin,
Experimental Quantum Cryptography,
J. Cryptology 5, 3 (1992).
11. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters,
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,
Phys. Rev. Lett. 70, 1895 (1993).
12. D. Boschi, S. Branca, F. De Martini, L. Hardy and S. Popescu,
Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and
Einstein-Podolsky-Rosen Channels,
Phys. Rev. Lett. 80, 1121 (1998).
13. H. Bennett and S. J. Wiesner,
Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,
Phys. Rev. Lett 69, 2881 (1992).
14. K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger,
Dense Coding in Experimental Quantum Communication,
Phys. Rev. Lett. 76, 4656 (1996).
15. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters,
Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels,
Phys. Rev. Lett. 76, 722–725 (1996).
16. C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K.
Wootters, Quantum nonlocality without entanglement,
Phys. Rev. A 59, 1070–1091 (1999).
17. A. Shimony, Degree of entanglement,
Ann. NY Acad. Sci. 755, 675 (1995).
18. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight,
Quantifying Entanglement,
Phys. Rev. Lett. 78, 2275 (1997).
19. T.-C. Wei and P. M. Goldbart,
Geometric measure of entanglement and applications to bipartite and multipartite quantum states,
Phys. Rev. A 68, 042307 (2003).
20. W. K. Wootters,
Entanglement of Formation of an Arbitrary State of Two Qubits,
Phys. Rev. Lett. 80, 2245 (1998).
21. B. M. Terhal and K. G. H. Vollbrecht,
Entanglement of Formation for Isotropic States,
Phys. Rev. Lett. 85, 2625 (2000).
22. L. Tamaryan, D. K. Park, and S. Tamaryan,
Analytic expressions for geometric measure of three-qubit states,
Phys. Rev. A 77, 022325 (2008).
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23. C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher,
Concentrating partial entanglement by local operations,
Phys.Rev.A 53, 2046(1996).
24. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin and W.K. Wootters,
Mixed-state entanglement and quantum error correction,
Phys.Rev.A 54, 3824(1996).
25. N. Seiberg and E. Witten,
Electric-Magnetic Duality, Monopole Condensation, and Confinement in N=2 Supersymmetric YangMills Theory,
Nucl.Phys.B 426, 19(1994).
26. A. Sen,
Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and
SL(2,Z) invariance in string theory,
Phys.Lett.B 29, 217(1994).
27. D.K. Park, S. Tamaryan, Y.-G. Miao, and H.J. W. Müller-Kirsten,
Tunneling of Born–Infeld strings to D2-branes,
Nucl.Phys.B 606, 84(2001).
28. D.K. Park, S. Tamaryan, and H. J.W. Müller-Kirsten,
D2-branes with magnetic flux in the presence of RR fields,
Nucl.Phys.B 635, 192(2002).
29. W. Dur, G. Vidal and J.I. Cirac,
Three qubits can be entangled in two inequivalent ways,
Phys.Rev.A 62, 062314(2000).
30. S. Tamaryan, A. Sudbery and L. Tamaryan,
Duality and the geometric measure of entanglement of general multiqubit W states, Phys.Rev.A 81,
052319 (2010).
31. L. Tamaryan, Z. Ohanyan and S. Tamaryan,
Universal behavior of the geometric entanglement measure of many-qubit W states, Phys.Rev.A 82,
022309 (2010).
32. S. Tamaryan,
Completely mixed state is a critical point for three-qubit entanglement,
Phys.Rev.A, submitted, arXiv: 1010.2848.
33. L. Tamaryan, D.K. Park, J.-W. Son, and S. Tamaryan,
Geometric Measure of Entanglement and Shared Quantum States,
Phys.Rev.A 78, 032304 (2008).
34. G. Rigolin,
Quantum teleportation of an arbitrary two-qubit state and its relation to multipartite entanglement,
Phys. Rev. A 71, 032303 (2005).
35. P. Agrawal and A. Pati,
Perfect teleportation and superdense coding with W states,
Phys.Rev.A 74, 062320 (2006).
36. E. Jung, M.-R. Hwang, M.-S. Kim, H. Kim, D. K. Park, J.-W. Son, and S. Tamaryan,
Greenberger-Horne-Zeilinger versus W states: Quantum teleportation through noisy channels,
Phys.Rev.A 78, 012312 (2008).
37. E. Jung, M.-R. Hwang, D. K. Park, J.-W. Son, and S. Tamaryan,
Mixed-state entanglement and quantum teleportation through noisy channels,
J.Phys.A: Math. Theor. 41, 385302 (2008).
38. S. Tamaryan, T.-C. Wei, and D.K. Park,
Maximally entangled three-qubit states via geometric measure of entanglement,
Phys.Rev.A 80, 052315 (2009).
39. E. Jung, M.-R. Hwang, H. Kim, M.-S. Kim, D.K. Park, J.-W. Son, and S. Tamaryan,
Reduced state uniquely defines the Groverian measure of the original pure state,
Phys. Rev. A 77, 062317 (2008).
40. N. Linden and W. K. Wootters,
The Parts Determine the Whole in a Generic Pure Quantum State,
Phys. Rev. Lett. 89, 277906 (2002).
41. H. Kim, M.-R. Hwang, E. Jung, and D.K. Park,
Difficulties in analytic computation for relative entropy of entanglement,
Phys. Rev. A 81, 052325 (2010)
42. C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and A. V. Thapliyal,
Exact and asymptotic measures of multipartite pure-state entanglement,
Phys.Rev.A 63, 012307 (2000).
7
Personnel Commitments (chart, total number of project participants, responsibilities
of each).
The team includes three researchers:
1. Principal investigator: Sayatnova Tamaryan, ANL [he is 54].
2. PhD-course student: Zaruhi Ohanyan, Department of Informatics and Applied
Mathematics, Yerevan state university [she is 24]
3. MS-course student:
Levon Tamaryan, Department of Physics, Yerevan state university
[he is 22]
Our team is the youngest team of A. Alikhanyan National Laboratory,
the average age of the members is 33 and nobody is at pension age.
Responsibilities of the participants are:
1. The principal investigator has the responsibility to create international scientific
collaborations, suggest objectives and clarify corresponding research methods.
2. Z. Ohanyan should carry out numerical optimizations over quantum protocols or states.
3. L. Tamaryan has the responsibility to compute analytically the quantum entanglement,
teleportation fidelity and related quantities.
Payments to Individual participants for each year
Participant
Salary (US $)
Annual Salary (US $)
1. Sayatnova Tamaryan
850
10 200
2. Zaruhi Ohanyan
650
7 800
3. Levon Tamaryan
600
7 200
2 100
25 200
Total
Refereed publications of the team within the years 2008-2010
8
1) L. Tamaryan, Z. Ohanyan, and S. Tamaryan,
Universal behavior of the geometric entanglement measure of many-qubit W states,
Physical Review A, 2010, 82, pp. 022309-022317.
2) S. Tamaryan, A. Sudbery, and L. Tamaryan,
Duality and the geometric measure of entanglement of general multiqubit W states,
Physical Review A, 2010, 81, pp. 052319-052322.
3) E. Jung, M.-R. Hwang, D.K. Park, and S. Tamaryan,
Three-party entanglement in tripartite teleportation scheme through noisy channels,
Quantum Information & Computation, 2010, 10, pp. 0377-0397.
4) S. Tamaryan, T.-C. Wei, and D.K. Park,
Maximally entangled three-qubit states via geometric measure of entanglement,
Physical Review A, 2009, 80, pp. 052315-052324.
5) L. Tamaryan, H. Kim, E. Jung, M.-R. Hwang , D.K. Park and S. Tamaryan,
Toward an understanding of entanglement for generalized n-qubit W-states,
Journal of Physics A:Math.Theor., 2009, 42, pp. 475303-475314.
6) E. Jung , M.-R. Hwang , D.K. Park , J.-W. Son and S. Tamaryan,
Mixed-state entanglement and quantum teleportation through noisy channels,
Journal of Physics A:Math.Theor., 2008, 41, pp. 385302-385312.
7) E. Jung , M.-R. Hwang , D.K. Park , L. Tamaryan, and S. Tamaryan,
Three-qubit Groverian measure,
Quantum Information & Computation, 2008, 8, pp.0951-0964.
8) L. Tamaryan, D.K. Park, J.-W. Son, and S. Tamaryan,
Geometric measure of entanglement and shared quantum states,
Physical Review A, 2008, 78, pp. 032304-032311.
9) E. Jung, M.-R. Hwang, D.K. Park, J.-W. Son, S. Tamaryan, and S.-K. Cha,
GHZ versus W states: Quantum teleportation through noisy channels
Physical Review A, 2008, 78, pp. 012312-012322.
10) L. Tamaryan, D.K. Park, and S. Tamaryan,
Analytic expressions for geometric measure of three-qubit states,
Physical Review A, 2008, 77, pp. 022325-022329.
11) E. Jung, M.-R. Hwang, H. Kim, M.-S. Kim, D.K. Park, J.-W. Son, and S. Tamaryan,
Reduced state uniquely defines the Groverian measure of the original pure state,
Physical Review A, 2008, 77, pp. 062317-062322.
9
Equipment
Equipment description
Cost (US $)
1.PC for the team, Intel (R) Core(TM) i5 CPU 750 - 2.67GHz
2.Printer for the team, HP LaserJet P2055D
http://hardware.am
3.Notebook for numerical optimization, 1.73 GHz Intel Core i7-740QM
4.Notebook for students, Intel Core i7-720QM 1.6GHz 6MB L2 Cache
http://iphone.am
1 600
400
Total
6 200
3 100
2 100
Materials
Materials description
Office supplies, filing cabinet,
stationery
Cost (US $)
2 500
Other Direct Costs
Direct cost description
Fee for reprints and colorful figures
Cost (US $)
2 000
Travel costs (US $)
CIS travel
None
International travel
1.Canada, University of British Columbia, $ 3 200
2.Britain, University of York
$ 2 200
3.Korea, Kyungnam University,
$ 2 600
10
Total
8 000
Technical Approach and Methodology
I.
Inverse duality of quantum entanglement
We will construct a duality such that slightly entangled GHZ states are mapped to highly
entangled W states and, conversely, highly entangled GHZ states are mapped to slightly
entangled W states. It is appropriate at this point to recall that the maximal product overlap g(ψ)
classifies quantum states as follows. The point g2(ψ) = 1/2 is the second critical value of the
entanglement and corresponding states |ψ are shared quantum states. Subsequently, the
states |ψ with g2(ψ) < 1/2 are highly entangled states and the states |ψ with g2(ψ) > 1/2 are
slightly entangled states.
The main advantage of the inverse duality is that it connects the maximal product
overlaps of a given GHZ-class state and its image W-class state and vice-versa. On the other
hand the maximal product overlap uniquely defines GM[19] and therefore if one knows GM of a
given W- or GHZ-class state, then one instantly finds GM of its image from the opposite class.
The duality transformation consists of two bijections. The first and the second bijections
map GHZ-class and W-class states to separable states respectively as follows
GHZ-states ↔ Separable states ↔ W-states
(1)
This is an analog of electromagnetic and strong-weak dualities and can shed new light
on the nature of quantum correlations. For instance, the duality transformation maps a subset
of pure GHZ-class states to the subset of mixed W-class states. Indeed, pure GHZ-class states
have five independent entanglement parameters, while pure W-class states have three of them
and therefore no bijection can connect them. But the full sets of GHZ- and W-class states
including both mixed and pure states have the same cardinality and it exists a one-to-one
correspondence (3) between these sets.
The duality (1) will be constructed as follows. We have shown that different highly
entangled pure W-states have different nearest product states [30]. This makes it possible to
map a pure W-state to its nearest product state and quickly obtain its geometric measure of
entanglement. More precisely, we have constructed two bijections. The first one creates a map
between highly entangled n-qubit W states and n-dimensional unit vectors xn. The second one
does the same between n-dimensional unit vectors and n-part product states. Thus we have
obtained a double map, or duality, as follows
W-states ↔ xn ↔ Product states
(2)
The main advantage of the map is that if one knows any of the three vectors, then one instantly
finds the other two. Hence we used this duality to find GM of W-class states.
The duality (2) can be extended into two directions: first, it can be extended to the mixed
W states and second, it can be extended to arbitrary GHZ states (work in progress).
Surprisingly, all of these dualities are parts of a general duality given by the formula (1).
11
II. Teleportation of unknown multipartite states
Quantum teleportation, or entanglement-assisted teleportation, is a process by which an
unknown quantum state can be transmitted exactly from one location to another, without the
state being transmitted through the intervening space. The original teleportation algorithm
[11,12] allows us to transmit an unknown one-qubit state from one location to another sending
2 bits of classical information. Next teleportation protocols were discovered that allow a sender
to faithfully teleport an arbitrary two-qubit state to a receiver sending him a 4 bit classical
message [34].
We will extend these teleportation protocols to arbitrary multi-qubit states including
mesoscopic and nanoscale pure states. The keystone idea of the extension is that the pure
states applicable for perfect teleportation should be shared quantum states and therefore their
maximal product overlap should has a critical value. Hence the perfect teleportation (dense
coding) is a quantum critical phenomenon in our approach and owing to this we can extend it to
the multipartite teleportation. First we will construct W states that can be used as a quantum
channel for a multipartite teleportation. Next we will explicitly show a protocol in which an
unknown multiqubit state is faithfully and deterministically teleported from a sender to a
receiver.
The duality (1) between W states and product states constructed in Ref.[30] revealed
two critical values for GM. The second critical value of GM is fundamental since all W states
with this value of GM can be used as a quantum channel for the perfect teleportation and
superdense coding. Besides, it guides us to engineer a quantum state for the multipartite
teleportation. Consider generalized W-states, which can be written
|W n = a1|100⋯0 + a2|0100⋯0 + ⋯ an|00⋯01.
(3)
Without loss of generality we assume that all coefficients are positive and real and 0 ≤ a1 ≤ a2
≤ ⋯ ≤ an. There is a critical value c of the largest coefficient an, i.e. a function of variables a1,
a2,⋯an-1, such that if an < c(a1, a2,⋯an-1), then the W state is highly entangled and if an > c(a1,
a2,⋯an-1),, then the W state is slightly entangled. The shared quantum states at an = c(a1,
a2,⋯an-1) are applicable for the perfect teleportation and dense coding. Setting n=2 and a3 =
c(a1, a2,) one gets exactly the original W state applicable for the perfect teleportation [35]. To
get a quantum state (up to normalization) capable to teleport an arbitrary n-qubit state one
simply uses the recurrence relation (unpublished yet!)
|Wk = |W k-1 ⊗ |0 + c(a1, a2,⋯ak-1) |00⋯01, k=4,5, ⋯ ,n+2
It remains to clarify the teleportation protocol, in particular, the measurement basis [36,37].
12
(4)
III. Quantum Communication Technology
The basic idea of quantum communication is to take advantage of the oddities of
quantum physics, like the uncertainty relation, the superposition principle and randomness.
One might argue that the laser, semiconductors, superconductivity, among others, are
technologies based on quantum physics. However, the big difference with quantum
communication - and more generally with quantum information science and technology - is that
it exploits quantum physics at the level of individual quanta.
Quantum teleportation allows for the transmission of quantum information to a distant
location despite the impossibility of measuring or broadcasting the information to be
transmitted. However, few-qubit teleportation schemes are not of great practical importance
since Holevo's bound establishes an upper bound on amount of information that can be
encoded in quantum states. More precisely, Holevo's theorem proves that n qubits can
represent only up to n classical (non-quantum encoded) bits [A.S. Holevo, Theor. Math. Phys.
14, 145 (1973)]. This is surprising, for two reasons: quantum computing is so often more
powerful than classical computing, that results which show it to be only as good or inferior to
conventional techniques are unusual, and because it takes 2n complex numbers to encode the
qubits which represent a mere n bits.
Thus in order to have a quantum channel that is capable to transmit n classical bits one
has to have a teleportation protocol that teleports unknown n-qubit states. We know precisely
the states that can be used as a quantum channel for multipartite teleportation (see equation
(4)), but we do not know the measurement basis and teleportation protocol.
Efficient long-distance quantum teleportation is crucial for quantum communication and
quantum networking schemes. Present experimental achievements are:
a) in August 2004 the distance of teleportation is increased to 600 meters using optical
fiber, [R. Ursin and et al, Nature 430, 849 (2004)],
b) the longest distance yet claimed to be achieved for quantum teleportation is 15 km
in May 2010 by Chinese scientists over free-space with an average of 89%
accuracy, [X.-M. Jin and et al, Nature Photonics 4, 376 - 381 (2010)].
We will rely on these schemes, but develop a quantum communication technology for
the teleportation of arbitrary multi-qubit states. First we need to fix the measurement basis and
teleportation protocol. Furthermore, real systems suffer from unwanted interactions with the
outside world. These unwanted interactions show up as noise in quantum information
processing systems. Besides, the perfect EPR state or another highly entangled pure states
can not be prepared initially due to noise.
Hence we need to elaborate a
multipartite
teleportation through noisy channels by mixed states. We have done it in the case of threequbits [37] and now we should extended it to the n-qubit case.
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The last remaining question is how much information can be transmitted using present
sources of coherent photon beams. Taking into account error corrections, we can assume that
the maximal amount is about seven thousand classical bits, but this estimation could be wrong.
We should examine the quantum state engineering process and related coherent photon beam
generation [31] and then estimate the number of auxiliary photons need to be consumed for
error correction.
Schematically the teleportation process can be illustrated as follows. First, the beam of
coherent photons is generated and then the sender and receiver share it.
Next the sender and receiver use local operations and classical communication to teleport an
unknown state.
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At the last stage the receiver restores the unknown state using the measurement results of the
sender.
The main advantages of the suggested quantum communication technology are:
1. The information stored in the quantum state can not be copied due to no-cloning
theorem [W.K. Wootters and W.H. Zurek, Nature (London) 299, 802 (1982)].
2. The information being transmitted can not be eavesdropped since the eavesdropper
should make measurements that tend unavoidably disturb the quantum state.
3. The teleportation process can not be suppressed since everybody between the sender
and receiver has no access to the information stored in the quantum state.
4. The suggested quantum technology is a long-distance(~15km) communication channel.
5. The channel has high capacity and, in principle, the capacity is unlimited which is the
contribution of our team.
Our basic research method is the duality which has revealed some important futures of
entangled states. Therefore, we will first complete tasks related directly to the duality and next
analyze multipartite teleportation and quantum communication technology.
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IV. The entanglement cost and the distillable entanglement of multipartite states
Bipartite systems have two basic operational entanglement measures: the entanglement
cost and the distillable entanglement [23,24] with the Einstein-Podolsky-Rosen state [1] |EPR
= (|↑↓ + |↓↑)/√2 emerging as the standard metric of entanglement. While the latter measure is
the rate at which copies of the maximally entangled EPR state can be concentrated from those
of a given state, the former is the rate at which copies of the maximally entangled EPR state
need to be consumed for the preparation of the given state.
The notions of distillable entanglement and entanglement cost are the fundamental
concepts of quantum information theory. However, they have not been properly generalized to
multipartite settings yet and this is one of main obstacles to elaborate a theory of multi-particle
entanglement. We will extend the entanglement cost and the distillable entanglement to
arbitrary multipartite states and compute them analytically for pure three-qubit states.
There is no simple and unique characterization of a maximally entangled state in
multipartite settings since the notion of being maximally entangled can depend on the choice of
entanglement measures. Then in order to generalize the entanglement cost and the distillable
entanglement to multipartite states one has to elucidate an important question of which pure
states can be regarded as the maximally entangled states. We gave a clear definition of
maximally entangled states as the states for which the maximal product overlap reaches to its
greatest lower bound [38]. Then we derived a single-parameter family of maximally entangled
three-qubit states by solving nonlinear eigenvalue equations for the maximal product overlap.
We have parameterized the maximally entanglement states with a gauge phase γ in
analogue to the two-qubit maximally entangled mixed states parameterized by the entropy.
Then the paradigmatic GHZ and W states emerge as extreme members in the family of
maximally entangled states. Now a pure state with the given gauge phase γ0 should be diluted
from the maximally entangled state with the same gauge phase γ0 (work in progress).
Unfortunately this is not the whole story. There are maximally entangled states whose gauge
phase is undefined, an example is the GHZ state |GHZ = (|000 + |111)/√2. One can ascribe
an arbitrary phase to the last term of the GHZ state and get an equally good maximally
entangled state. All of these maximally entangled states with undefined phases should
participate in the entanglement dilution protocol, as one can ascribe the gauge phase γ0 to
these states. Then one should dilute a given state from a finite subset of the maximally
entangled states and this is just the concept of minimal reversible entanglement generating
set(MREGS) [42].
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V. Relative entropy of entanglement of pure three-qubit states
The three-qubit system is important in the sense that it is the simplest system which gives a
nontrivial effect in the entanglement. Unlike to bipartite systems, tripartite systems have
inequivalent entanglement classes and owing to this the entanglement has critical values.
Thus, we should understand the general properties of the entanglement in this system as much
as possible to go further to more complicated higher qubit systems.
The computation of the relative entropy of entanglement gives a nonlinear eigenvalue
problem which, except in rare cases, does not allow complete analytical solutions. Recently the
idea was suggested that the nonlinear eigenproblem can be reduced to the linear eigenproblem
for the case of three-qubit pure states [39]. The idea is based on a theorem stating that any
reduced (n-1)-qubit state uniquely determines an arbitrary entanglement measure of the
original n-qubit pure state [39, 40]. This means that two qubit mixed states can be used to
calculate the relative entropy of entanglement of three-qubit pure states and this will be fully
addressed in this proposal.
The method gives two algebraic equations of degree 6 defining the relative entropy of
entanglement. Thus the difficult problem of entanglement calculations is reduced to algebraic
equation root finding. The equations contain valuable information, are good bases for the
numerical calculations, and may test numerical calculations based on other numerical
techniques. Furthermore, the method allows one to find the nearest separable states for threequbit states of most interest and obtain analytic expressions for their relative entropy of
entanglement.
At the last stage we will analyze the relative entropy of entanglement at the critical
points of entanglement discovered in [30]. The term critical point has a mathematical and a
physical justifications. The mathematical justification is the following. The geometric
entanglement measure of quadrilateral three-qubit [33] and general W [30] states have been
computed analytically and the answers show the gradient of the measure has a jump at these
points. Hence they are critical points. The physical justification is that at the edge of the region
of possible values of a quartic polynomial invariant the entanglement of the state is absolutely
independent of the remaining five polynomial invariants and then the state acquires an ability to
be a quantum channel for the perfect teleportation and dense coding.
Our aim is to clarify:
a) whether or not these features of the criticality are valid for the relative entropy of
entanglement too and
b) does the relative entropy entanglement has its own critical value or is the critical value
of GM found in [30] unique for all entanglement measures?
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