marcos curty 1 2 coauthors tobias moroder 2 3 and norbert l tkenhaus 2 3 n.
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  1. 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

  2. Overview • Quantum Key Distribution (QKD) • Precondition for secure QKD (Two-way & One-way) • Witness Operators (Two-way & One-way QKD) • Semidefinite Programming • Evaluation

  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

  4. 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)

  5. 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)

  6. 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)

  7. 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)

  8. 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)

  9. 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)

  10. 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

  11. 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?

  12. 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?

  13. 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

  14. 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)

  15. 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.

  16. 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

  17. 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

  18. 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)

  19. • 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)

  20. 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).

  21. 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)

  22. 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).