Marcos curty 1 2 coauthors tobias moroder 2 3 and norbert l tkenhaus 2 3
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Marcos Curty 1,2 Coauthors: Tobias Moroder 2,3 , and Norbert Lütkenhaus 2,3 PowerPoint PPT Presentation


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On One-way and Two-way Classical Post-Processing Quantum Key Distribution. Marcos Curty 1,2 Coauthors: Tobias Moroder 2,3 , and Norbert Lütkenhaus 2,3. Center for Quantum Information and Quantum Control (CQIQC), University of Toronto Institute for Quantum Computing, University of Waterloo

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Marcos Curty 1,2 Coauthors: Tobias Moroder 2,3 , and Norbert Lütkenhaus 2,3

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On One-way and Two-way Classical Post-Processing Quantum Key Distribution

Marcos Curty1,2

Coauthors:Tobias Moroder2,3, and Norbert Lütkenhaus2,3

  • Center for Quantum Information and Quantum Control (CQIQC), University of Toronto

  • Institute for Quantum Computing, University of Waterloo

  • Max-Plank-Forschungsgruppe, Institut für Optik, Information und Photonik, Universität Erlangen-Nürnberg


Overview

  • Quantum Key Distribution (QKD)

  • Precondition for secure QKD (Two-way & One-way)

  • Witness Operators (Two-way & One-way QKD)

  • Semidefinite Programming

  • Evaluation


Ai

Ai

Bj

AiAi

Pr(Ai,Bj)=Pr(Ai)Tr(Bj )

Mathematical Model

AB

Bj

Ai

AB

Pr(Ai,Bj)=Tr(Ai Bj )

AB=i Pr(Ai)1/2AiAi with AB= AB

Ai 1

A= TrB(AB)

Reduced density matrix of Alice fixed Add:

Quantum Key Distribution (QKD)

Phase I: Physical Set-Up


Two-way

Pr(Ai,Bj)

Secret key

Authenticated Classical Channel

Quantum Key Distribution (QKD)

Phase II: Classical Communication Protocol

  • Advantage distillation (e.g. announcement of bases in BB84 protocol)

  • Error Correction ( Alice and Bob share the same key)

  • Privacy Amplification ( generates secret key shared by Alice and Bob)


One-way (Reverse Reconciliation: RR)

Pr(Ai,Bj)

Secret key

Authenticated Classical Channel

Quantum Key Distribution (QKD)

Phase II: Classical Communication Protocol

  • Advantage distillation (e.g. announcement of bases in BB84 protocol)

  • Error Correction ( Alice and Bob share the same key)

  • Privacy Amplification ( generates secret key shared by Alice and Bob)


One-way (Direct communication: DC)

Pr(Ai,Bj)

Secret key

Authenticated Classical Channel

Quantum Key Distribution (QKD)

Phase II: Classical Communication Protocol

  • Advantage distillation (e.g. announcement of bases in BB84 protocol)

  • Error Correction ( Alice and Bob share the same key)

  • Privacy Amplification ( generates secret key shared by Alice and Bob)


Talk: T. Moroder

secret bits

per signal

Not secure (proven)

Protocol independent

Regime of Hope

Talk: G. O. Myhr

This talk

secure

(proven)

protocol

Distance (channel model)

Which type of correlations Pr(Ai,Bj) are useful for QKD?

Quantum Key Distribution (QKD)


AB

Ai

Pr(Ai,Bj)

Bj

ABseparable

No Key

AB is separable if

AB=i pi |aiai|A|bibi|B

MC, M. Lewenstein and N. Lütkenhaus, Phys. Rev. Lett. 92,217903 (2004)

Precondition for Secure QKD

Theorem (Two-way QKD)


AB

Ai

Pr(Ai,Bj)

Bj

ABhas a symmetric extension to two-copies of system B

(A), then the secret key rate for direct communication

(reverse reconciliation) vanishes.

T. Moroder, MC and N. Lütkenhaus, quant-ph/0603270.

Precondition for Secure QKD

Theorem (One-way QKD)


AB

AB

A

B

A

B

TrE(ABE)= AB

ABE

E

E

A

B

TrB(ABE)= AE = AB

AB

E

Precondition for Secure QKD

AB with symmetric extension to two copies of system B


TrWAB < 0

Witness Operators

TrWAB 0  ABcomp.with separable

Accesible witnesses:W = ij cij AiBj

  • restricted knowledge

Optimal Wopt

Wopt

verifiable

entangled

TrWAB = ij cij P(Ai,Bj )

compatible

with sep.

AB

W

MC, M. Lewenstein and N. Lütkenhaus, Phys. Rev. Lett. 92,217903 (2004)

MC, O. Gühne, N. Lewenstein, N. Lütkemhaus, Phys. Rev. A 71, 022306 (2005)

Witness Operators (Two-way QKD)

ABseparable?


Witness Operators

  • restricted knowledge

Without

symmetric

extension

compatible

with symmetric

extension.

TrWAB < 0

TrWAB 0  ABcomp.with symmetric extension

AB

Accesible witnesses:W = ij cij AiBj

Wopt

TrWAB = ij cij P(Ai,Bj )

T. Moroder, MC and N. Lütkenhaus, quant_ph/0603270.

Witness Operators (One-way QKD)

ABsymmetric extension?


Pr(Ai,Bj)

|1

|1

|0

A\B

0 1 01

0

1

0

1

0.07987 0.04516 0.00913 0.11591

0.04508 0.07986 0.11593 0.00901

0.11599 0.00909 0.08001 0.04507

0.00897 0.11593 0.04505 0.07985

|0

Uses two mutually unbiased bases:

e.g. X,Z direction in Bloch sphere

Error Rate: 36 %

W4 = 1/2(|ee| + |ee|)

Systematic Search

| e=cos(X)|00+sin(X)(cos(Y)|01+sin(Y)(cos(Z)|10+sin(Z)|11))

MC, M. Lewenstein and N. Lütkenhaus, Phys. Rev. Lett. 92,217903 (2004)

Witness Operators (Two-way QKD)

Evaluation: 4-state QKD protocol


Witness Operators (Two-way QKD)

Evaluation: 4-state QKD protocol

(only parameter combinations leading to negative expectation values are marked)

TrWAB = ij cij P(Ai,Bj )

MC, O. Gühne, N. Lewenstein, N. Lütkemhaus, Phys. Rev. A 71, 022306 (2005)

MC, O. Gühne, N. Lewenstein, N. Lütkemhaus, Proc. SPIE Int. Soc. Opt. Eng.

5631, 9-19 (2005).

J. Eisert, P. Hyllus, O. Gühne, MC, Phys. Rev. A 70, 062317 (2004).

Other QKD protocols (including

higher dimensional QKD schemes)


Witness Operators (Two-way and One-way QKD)

Advantages: Witnesses operators

• One witness: Sufficient condition as a first step towards the demonstration of the feasibility of a particular experimental implementation of QKD. This criterion is independent of any chosen communication protocol in Phase II.

• All witnesses: Systematic search for quantum correlations (or symmetric extensions) for a given QKD setup. Ideally the main goal is to obtain a compact description of a minimal verification set of witnesses (Necessary-and Sufficient).

Disadvantages: Witnesses operators

• How to find them?:To find a minimal verification set of EWs, even for qubit-based QKD schemes, is not always an easy task, and it seems to require a whole independent analysis for each protocol.

• Too many tests:To guarantee that no secret key can be obtained from the observed data it is necessary to test all the members of the minimal verification set.


Primal problem

minimise cTx

subject to F0+i xi Fi ≥ 0

with x=(x1, ..., xt)T the objective variable, c is fixed by the optimisation problem, and the matrices Fi are Hermitian

Equivalent class of states S

S = {AB such as Tr(Ai  Bj AB) = Pr(Ai,Bj)  i,j}

Semidefinite Programming (SDP)

SDPs can be

efficiently solved

Qubit-based QKD (with losses): AB  H2H3


SDP

Feasibility problem

c = 0

AB  S with AB  0

Bj

AB

Ai

minimise 0

subject to AB(x)  0

AB(x)  0

AB(x)  S

Pr(Ai,Bj)

No Key

MC, T. Moroder, and N. Lütkenhaus, in preparation (2006)

Semidefinite Programming (SDP)

Two-way QKD


SDP: One-way QKD

minimise 0

subject to AB(x)  SPABA’(x)P = ABA’(x)

ABA’(x)  0 TrA’[ABA’(x)] = AB(x)

with P the swap operator: P|ijkABA’ = |kjiABA’

MC, T. Moroder, and N. Lütkenhaus, in preparation (2006)

Dual problem

maximise -Tr(F0 Z)

subject to Z ≥ 0

Tr(Fi Z) = ci for all i

where the Hermitian Z is the objective variable

Semidefinite Programming (SDP)

Dual problem (one way & two-way)  Witness operator optimal for Pr(Ai,Bj)


• Channel Model:

AB = (1-p)[(1-e)|AB|+e/2 A1B] + p A|vacBvac|

p: probability Bob receives the vacuum state |vacB

e: depolarizing rate

1B: 1B- |vacBvac|

Evaluation

• We need experimental data Pr(Ai,Bj)


Six-state protocol:

|1

Alice and Bob:

|1

|1

|0

|0

|0

Bruß, Phys. Rev. Lett. 81, 3018 (1998).

Four-state protocol:

Alice and Bob:

|1

|1

|0

|0

C.H. Bennett and G. Brassard, Proc. IEEE Int. Conf. On Computers, System and Signal Processing, 175 (1984).

Evaluation

QBER: 33 %

QBER: 16.66 %

H. Bechmann-Pasquinucci, and

N. Gisin, Phys. Rev. A 59, 4238

(1999).

QBER: 25 %

QBER: 14.6 %

C. A. Fuchs, N. Gisin, R. B. Griffiths, C.-S. Niu, and

A. Peres, Phys. Rev. A 56, 1163 (1997); J. I. Cirac, and N. Gisin, Phys. Lett. A 229, 1 (1997).


Two-state protocol:

Alice:

Bob:

|0 = |0+|1

B0 = 1/(22)|11|

B1 = 1/(22)|00|

B? = |00|+|11|-B0-B1

Bvac = |vacvac|

|1 = |0-|1

C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992).

Four-plus-two-state protocol:

Like 2 two-state protocols:

|1

|1

|0

|0

B. Huttner, N. Imoto, N. Gisin, and T. Mor, Phys. Rev. A 51, 1863 (1995).

Evaluation

Limit USD

p1-22

e=0

Inflexion point

e constant

p=1-22

(USD)

Other QKD protocols 

MC, T. Moroder, and N. Lütkenhaus, in preparation (2006)


Summary

  • Interface Physics – Computer Science:Classical Correlated Data with a Promise

  • Necessary condition for secure QKD(Two-way & One-way).

  • Relevance for experiments: show the presence of entanglement (states without symmetric

  • extension)

    • No need to enter details of classical communication protocols

    • Prevent oversights in preliminary analysis

    • One properly constructed proof suffices

  • Evaluation: Semidefinite programming (qubit-based QKD protocols in the presence of loss).

  • Task for Theory: Develop practical tools for realistic experiments ( for given measurements).


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