Marcos curty 1 2 coauthors tobias moroder 2 3 and norbert l tkenhaus 2 3
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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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Marcos curty 1 2 coauthors tobias moroder 2 3 and norbert l tkenhaus 2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

Overview

  • Quantum Key Distribution (QKD)

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

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

  • Semidefinite Programming

  • Evaluation


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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?


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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?


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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.


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

• 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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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).


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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)


Marcos curty1 2 coauthors tobias moroder2 3 and norbert l tkenhaus2 3

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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