Reliable numerical key rates for quantum key distribution

Adam Winick, Norbert Lütkenhaus, and Patrick J. Coles

Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, N2L3G1 Waterloo, Ontario, Canada

In this work, we present a reliable, efficient, and tight numerical method for calculating key rates for finite-dimensional quantum key distribution (QKD) protocols. We illustrate our approach by finding higher key rates than those previously reported in the literature for several interesting scenarios (e.g., the Trojan-horse attack and the phase-coherent BB84 protocol). Our method will ultimately improve our ability to automate key rate calculations and, hence, to develop a user-friendly software package that could be used widely by QKD researchers.

The main theoretical challenge in quantum key distribution (QKD) is to evaluate the performance of a given protocol, as quantified by the key rate (bits of secret key per transmitted quantum signal). Analytical methods for key rate calculations have made significant progress for specific
protocols, but often tend not to be robust to device imperfections or changes in protocol structure. Numerical methods naturally have more robustness, but new issues arise with numerics such as the reliability and efficiency (i.e., computation time) of the calculation. In this work, we present a reliable, efficient, and tight numerical method for calculating key rates for finite-dimensional QKD protocols. Due to its tightness, our method gives higher key rates than literature methods for several
technologically important scenarios, for example, for the Trojan-horse attack and for the phase-coherent BB84 protocol. Our method will ultimately allow researchers to automate key rate calculations and hence to develop user-friendly software for QKD security analysis.

► BibTeX data

► References

[1] Campagna, M. et al. Quantum Safe Cryptography and Security (European Telecommunications Standards Institute, 2015).

[2] Wyner, A. D. The Wire-Tap Channel. Bell System Technical Journal 54, 1355-1387 (1975). URL https:/​/​​10.1002/​j.1538-7305.1975.tb02040.x.

[3] Scarani, V. et al. The security of practical quantum key distribution. Reviews of Modern Physics 81, 1301-1350 (2009). URL https:/​/​​10.1103/​RevModPhys.81.1301.

[4] Lo, H.-K., Curty, M. & Tamaki, K. Secure quantum key distribution. Nature Photonics 8, 595-604 (2014). URL https:/​/​​10.1038/​nphoton.2014.149.

[5] Xin, H. Chinese Academy Takes Space Under Its Wing. Science 332, 904-904 (2011). URL https:/​/​​10.1126/​science.332.6032.904.

[6] Peev, M. et al. The SECOQC quantum key distribution network in Vienna. New Journal of Physics 11, 075001 (2009). URL https:/​/​​10.1088/​1367-2630/​11/​7/​075001.

[7] Sasaki, M. et al. Field test of quantum key distribution in the Tokyo QKD Network. Optics Express 19, 10387-10409 (2011). URL https:/​/​​10.1364/​OE.19.010387.

[8] Wang, S. et al. Field and long-term demonstration of a wide area quantum key distribution network. Optics Express 22, 21739 (2014). URL https:/​/​​10.1364/​OE.22.021739.

[9] Renner, R. Security of Quantum Key Distribution. Ph.D. thesis, ETH Zurich (2005). URL http:/​/​​abs/​quant-ph/​0512258.

[10] Renner, R., Gisin, N. & Kraus, B. Information-theoretic security proof for quantum-key-distribution protocols. Physical Review A 72, 012332 (2005). URL https:/​/​​10.1103/​PhysRevA.72.012332.

[11] Watanabe, S., Matsumoto, R. & Uyematsu, T. Tomography increases key rates of quantum-key-distribution protocols. Physical Review A 78, 042316 (2008). URL https:/​/​​10.1103/​PhysRevA.78.042316.

[12] Matsumoto, R. Improved asymptotic key rate of the B92 protocol. IEEE International Symposium on Information Theory - Proceedings 351-353 (2013). URL https:/​/​​10.1109/​ISIT.2013.6620246.

[13] Vakhitov, A., Makarov, V. & Hjelme, D. R. Large pulse attack as a method of conventional optical eavesdropping in quantum cryptography. Journal of Modern Optics 48, 2023-2038 (2001). URL https:/​/​​10.1080/​09500340108240904.

[14] Gisin, N., Fasel, S., Kraus, B., Zbinden, H. & Ribordy, G. Trojan-horse attacks on quantum-key-distribution systems. Physical Review A 73, 022320 (2006). URL https:/​/​​10.1103/​PhysRevA.73.022320.

[15] Lucamarini, M. et al. Practical Security Bounds Against the Trojan-Horse Attack in Quantum Key Distribution. Physical Review X 5, 031030 (2015). URL https:/​/​​10.1103/​PhysRevX.5.031030.

[16] Huttner, B., Imoto, N., Gisin, N. & Mor, T. Quantum cryptography with coherent states. Physical Review A 51, 1863-1869 (1995). URL https:/​/​​10.1103/​PhysRevA.51.1863.

[17] Lo, H. & Preskill, J. Security of quantum key distribution using weak coherent states with nonrandom phases. Quantum Information and Computation 7, 431-458 (2007). URL http:/​/​​abs/​quant-ph/​0610203.

[18] Fung, C. C.-H. F., Tamaki, K., Qi, B., Lo, H.-K. H. & Ma, X. Security proof of quantum key distribution with detection efficiency mismatch. Quantum Inf. Comput. 9, 0131-0165 (2009). URL http:/​/​​citation.cfm?id=2021264.

[19] Coles, P. J., Metodiev, E. M. & Lütkenhaus, N. Numerical approach for unstructured quantum key distribution. Nature Communications 7, 11712 (2016). URL https:/​/​​10.1038/​ncomms11712.

[20] Boyd, S. & Vandenberghe, L. Convex Optimization (Cambridge University Press, 2004). URL http:/​/​​ boyd/​cvxbook/​.

[21] Devetak, I. & Winter, A. Distillation of secret key and entanglement from quantum states. Proceedings of the Royal Society A 461, 207-235 (2005). URL https:/​/​​10.1098/​rspa.2004.1372.

[22] Renner, R. Symmetry of large physical systems implies independence of subsystems. Nature Physics 3, 645-649 (2007). URL https:/​/​​10.1038/​nphys684.

[23] Petersen, K. B. & Pedersen, M. S. The Matrix Cookbook (2012). URL http:/​/​​pubdb/​p.php?3274.

[24] Al-Mohy, A. H. & Higham, N. J. Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm. SIAM Journal on Scientific Computing 34, C153-C169 (2012). URL https:/​/​​10.1137/​110852553.

[25] Frank, M. & Wolfe, P. An algorithm for quadratic programming. Naval Research Logistics Quarterly 3, 95-110 (1956). URL https:/​/​​10.1002/​nav.3800030109.

[26] Bennett, C. H. & Brassard, G. Quantum cryptography: public key distribution and coin tossing. In International Conference on Computers, Systems & Signal Processing, Bangalore, India, 175-179 (1984). URL https:/​/​​10.1016/​j.tcs.2011.08.039.

[27] Bennett, C. H. Quantum cryptography using any two nonorthogonal states. Physical Review Letters 68, 3121-3124 (1992). URL https:/​/​​10.1103/​PhysRevLett.68.3121.

[28] Scarani, V., Acín, A., Ribordy, G. & Gisin, N. Quantum Cryptography Protocols Robust against Photon Number Splitting Attacks for Weak Laser Pulse Implementations. Physical Review Letters 92, 057901 (2004). URL https:/​/​​10.1103/​PhysRevLett.92.057901.

[29] Bruß, D. Optimal Eavesdropping in Quantum Cryptography with Six States. Physical Review Letters 81, 3018-3021 (1998). URL https:/​/​​10.1103/​PhysRevLett.81.3018.

[30] Lo, H.-K., Ma, X. & Chen, K. Decoy State Quantum Key Distribution. Physical Review Letters 94, 230504 (2005). URL https:/​/​​10.1103/​PhysRevLett.94.230504.

[31] Lo, H.-K., Curty, M. & Qi, B. Measurement-Device-Independent Quantum Key Distribution. Physical Review Letters 108, 130503 (2012). URL https:/​/​​10.1103/​PhysRevLett.108.130503.

[32] Stacey, W., Annabestani, R., Ma, X. & Lütkenhaus, N. Security of quantum key distribution using a simplified trusted relay. Physical Review A 91, 012338 (2015). URL https:/​/​​10.1103/​PhysRevA.91.012338.

[33] Bennett, C., Brassard, G. & Mermin, N. Quantum cryptography without Bell's theorem. Physical Review Letters 68, 557-559 (1992). URL https:/​/​​10.1103/​PhysRevLett.68.557.

[34] Ferenczi, A. & Lütkenhaus, N. Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning. Physical Review A 85, 052310 (2012). URL https:/​/​​10.1103/​PhysRevA.85.052310.

[35] Coles, P. J. Unification of different views of decoherence and discord. Physical Review A 85, 042103 (2012). URL https:/​/​​10.1103/​PhysRevA.85.042103.

[36] Sajeed, S. et al. Security loophole in free-space quantum key distribution due to spatial-mode detector-efficiency mismatch. Physical Review A 91, 062301 (2015). URL https:/​/​​10.1103/​PhysRevA.91.062301.

[37] Lydersen, L. et al. Hacking commercial quantum cryptography systems by tailored bright illumination. Nature Photonics 4, 686-689 (2010). URL http:/​/​​10.1038/​nphoton.2010.214.

[38] Scarani, V. & Renner, R. Quantum Cryptography with Finite Resources: Unconditional Security Bound for Discrete-Variable Protocols with One-Way Postprocessing. Physical Review Letters 100, 200501 (2008). URL https:/​/​​10.1103/​PhysRevLett.100.200501.

[39] Sano, Y., Matsumoto, R. & Uyematsu, T. Secure key rate of the BB84 protocol using finite sample bits. IEEE International Symposium on Information Theory - Proceedings 43, 2677-2681 (2010). URL https:/​/​​10.1109/​ISIT.2010.5513653.

[40] Tomamichel, M., Ci, C., Lim, W., Gisin, N. & Renner, R. Tight finite-key analysis for quantum cryptography. Nature Communications 3, 634 (2012). URL http:/​/​​10.1038/​ncomms1631.

[41] Coles, P. J., Kaniewski, J. & Wehner, S. Equivalence of wave-particle duality to entropic uncertainty. Nature Communications 5, 5814 (2014). URL http:/​/​​10.1038/​ncomms6814.

[42] Nielsen, M. A. & Chuang, I. Quantum Computation and Quantum Information (Cambridge University Press, 2000).

[43] Lo, H.-K., Chau, H. & Ardehali, M. Efficient Quantum Key Distribution Scheme and a Proof of Its Unconditional Security. Journal of Cryptology 18, 133-165 (2005). URL https:/​/​​10.1007/​s00145-004-0142-y.

[44] Gittsovich, O. et al. Squashing model for detectors and applications to quantum-key-distribution protocols. Physical Review A 89, 012325 (2014). URL https:/​/​​10.1103/​PhysRevA.89.012325.

► Cited by (beta)

Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.