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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 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 AiAi Pr(Ai,Bj)=Pr(Ai)Tr(Bj ) Mathematical Model AB Bj Ai AB Pr(Ai,Bj)=Tr(Ai Bj ) AB=i Pr(Ai)1/2AiAi 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 |aiai|A|bibi|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**TrWAB < 0**Witness Operators TrWAB 0 ABcomp.with separable Accesible witnesses:W = ij cij AiBj • restricted knowledge Optimal Wopt Wopt verifiable entangled TrWAB = 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. TrWAB < 0 TrWAB 0 ABcomp.with symmetric extension AB • Accesible witnesses:W = ij cij AiBj Wopt TrWAB = 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(|ee| + |ee|) 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) TrWAB = 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 H2H3**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) SPABA’(x)P = ABA’(x) ABA’(x) 0 TrA’[ABA’(x)] = AB(x) with P the swap operator: P|ijkABA’ = |kjiABA’ 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 A1B] + p A|vacBvac| p: probability Bob receives the vacuum state |vacB e: depolarizing rate 1B: 1B- |vacBvac| 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/(22)|11| B1 = 1/(22)|00| B? = |00|+|11|-B0-B1 Bvac = |vacvac| |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 p1-22 e=0 Inflexion point e 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).