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Fundamental finite key limits for one-way information reconciliation in quantum key distribution

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Abstract

The security of quantum key distribution protocols is guaranteed by the laws of quantum mechanics. However, a precise analysis of the security properties requires tools from both classical cryptography and information theory. Here, we employ recent results in non-asymptotic classical information theory to show that one-way information reconciliation imposes fundamental limitations on the amount of secret key that can be extracted in the finite key regime. In particular, we find that an often used approximation for the information leakage during information reconciliation is not generally valid. We propose an improved approximation that takes into account finite key effects and numerically test it against codes for two probability distributions, that we call binary–binary and binary–Gaussian, that typically appear in quantum key distribution protocols.

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Notes

  1. Consider, for example, BB84 with asymmetric basis choice [25] on a channel with quantum bit error rate Q. Here, contributions (1) and (2) vanish asymptotically while contributions (3) and (4) converge to h(Q).

  2. Recent works analyzing the finite block length behavior using this approximation include [1, 5, 7, 17, 24, 35, 43].

  3. We here apply Theorem 1 to distributions that are continuous in Y. Note that the proofs leading to Theorem 1 can easily be generalized to this setting.

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Acknowledgements

MT thanks N. Beaudry, S. Bratzik, F. Furrer, M. Hayashi, C.C.W. Lim, and V.Y.F. Tan for helpful comments and pointers to related work. MT is supported by an Australian Research Council Discovery Early Career Researcher Award (DECRA) fellowship. JM has been funded by the Spanish Ministry of Economy and Competitiveness through project Continuous Variables for Quantum Communications (CVQuCo), TEC2015-70406-R. CP has been funded by the Vienna Science and Technology Fund (WWTF) through project ICT10-067 (HiPANQ). DE was supported via STW and the NWO Vidi grant “Large quantum networks from small quantum devices”.

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Authors and Affiliations

  1. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, NSW, 2007, Australia

    Marco Tomamichel

  2. Center for Computational Simulation, Universidad Politecnica de Madrid, 28660, Boadilla del Monte, Spain

    Jesus Martinez-Mateo

  3. Digital Safety & Security Department, AIT Austrian Institute of Technology, Donau-City-Straße 1, 1220, Vienna, Austria

    Christoph Pacher

  4. QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA, Delft, The Netherlands

    David Elkouss

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  1. Marco Tomamichel
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Correspondence to Marco Tomamichel.

Additional information

Part of these results without the technical derivations were published in the proceedings of the International Symposium on Information Theory, Honolulu (2014) [44].

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Tomamichel, M., Martinez-Mateo, J., Pacher, C. et al. Fundamental finite key limits for one-way information reconciliation in quantum key distribution. Quantum Inf Process 16, 280 (2017). https://doi.org/10.1007/s11128-017-1709-5

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