Marcos Curty 1,2 Coauthors: Tobias Moroder 2,3, and Norbert Lütkenhaus 2,3 1.Center for Quantum Information and Quantum Control (CQIQC), University of

Marcos Curty1,2

Coauthors: Tobias Moroder2,3, and Norbert Lütkenhaus2,3 1. Center for Quantum Information and Quantum Control (CQIQC), University of Toronto2. Institute for Quantum Computing, University of Waterloo3. Max-Plank-Forschungsgruppe, Institut für Optik, Information und Photonik, Universität

Erlangen-Nürnberg

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

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

way)• Witness Operators (Two-way & One-way QKD)• Semidefinite Programming• Evaluation

Overview

Quantum Key Distribution (QKD)

Phase I: Physical Set-Up

Mathematical Model

AiBj

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

AB=i Pr(Ai)1/2AiAi with AB= AB

AB

Bj

Ai

Ai

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

Reduced density matrix of Alice fixed Add: A= TrB(AB) Ai 1


Phase II: Classical Communication Protocol

Pr(Ai,Bj) Secret key

• 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)

Authenticated Classical Channel

Two-way






One-way (Reverse Reconciliation: RR)






One-way (Direct communication: DC)


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

secret bitsper signal

Distance (channel model)

Not secure (proven) Protocol independent

Regime of Hope

secure(proven)protocol

Talk: T. Moroder

This talk

Talk: G. O. Myhr

Precondition for Secure QKD

Theorem (Two-way QKD)

AB

Pr(Ai,Bj)

AB separable No Key

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

Ai Bj

AB is separable if AB = i pi |aiai|A|bibi|B


Theorem (One-way QKD)

AB

Pr(Ai,Bj)

AB has 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.

Ai Bj


AB with symmetric extension to two copies of system B

AB

TrE(ABE)= AB

ABE

A B

E

A B

E

AB

A B

E

TrB(ABE)= AE = AB AB

Witness Operators (Two-way QKD)

AB separable?

TrWAB < 0 TrWAB 0 AB comp.with

separable

Witness Operators

TrWAB = ij cij P(Ai,Bj )

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)

restricted knowledge

compatiblewith sep.

verifiableentangled

• AB

W

Accesible witnesses: W = ij cij Ai Bj

Optimal Wopt

Wopt

Witness Operators (One-way QKD)

AB symmetric extension?

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


TrWAB < 0 TrWAB 0 AB comp. with symmetric extension

restricted knowledge

compatiblewith symmetric

extension.

Without symmetric extension

• AB

Wopt

Witness Operators

Accesible witnesses: W = ij cij Ai Bj


Evaluation: 4-state QKD protocol

Uses two mutually unbiased bases:e.g. X,Z direction in Bloch sphere

|0|1|1

|0

Error Rate: 36 %

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

0101

0 1 0 1A\B

Pr(Ai,Bj)

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

Systematic Search

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

W4 = 1/2(|ee| + |ee|)


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

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).

Evaluation: 4-state QKD protocol


Other QKD protocols (including higher dimensional QKD schemes)

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

• 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).

Advantages: Witnesses operators

Disadvantages: Witnesses operators

• 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.

• 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.

Semidefinite Programming (SDP)

Primal problem

minimise cTxsubject 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

SDPs can be efficiently solved

Equivalent class of states S

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

Qubit-based QKD (with losses): AB H2H3

Semidefinite Programming (SDP)

Two-way QKD

AB S with AB 0 No Key

AB

Pr(Ai,Bj) Ai Bj

SDP

Feasibility problemc = 0

minimise 0subject to AB(x) 0 AB

(x) 0

AB(x) SS

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)SDP: One-way QKD

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

minimise 0subject to AB(x) SS PABA’(x)P = ABA’(x)

ABA’(x) 0 TrA’[ABA’(x)] = AB(x)

with P the swap operator: P|ijkABA’ = |kjiABA’

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

Evaluation

• We need experimental data Pr(Ai,Bj)

• Channel Model:

AB = (1-p)[(1-e)|AB|+e/2 A1B] + p A|vacBvac|

p: probability Bob receives the vacuum state |vacB e: depolarizing rate1B: 1B- |vacBvac|

Evaluation

Six-state protocol:

|0

|00

|1|1

|0

|11Alice and Bob:

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

Four-state protocol:

|0

|00

|1

|11Alice and Bob:

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

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

Evaluation Two-state protocol:

Alice:

|0 = |0+|1

|1 = |0-|1 C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992).

Bob:

B0 = 1/(22)|11

|

B1 = 1/(22)|00

|

B? = |00|+|11|-B0-B1

Bvac = |vacvac|

Limit USD p1-22

e=0Four-plus-two-state protocol:

|0|1|1

|0

Like 2 two-state protocols:

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

Inflexion pointe constant

p=1-22

(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).

Documents

Marcos Curty 1,2 Coauthors: Tobias Moroder 2,3, and Norbert Lütkenhaus 2,3 1.Center for Quantum Information and Quantum Control (CQIQC), University of