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Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

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Abstract

A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a “sandwiched” Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Center for Computation and Technology, Hearne Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Mark M. Wilde

  2. ICREA, Informació i Fenomens, Quàntics, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Andreas Winter

  3. Física Teòrica, Informació i Fenomens, Quàntics, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Andreas Winter & Dong Yang

  4. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Andreas Winter

  5. Laboratory for Quantum Information, China Jiliang University, Hangzhou, 310018, Zhejiang, China

    Dong Yang

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  1. Mark M. Wilde
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  2. Andreas Winter
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  3. Dong Yang
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Correspondence to Mark M. Wilde.

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Communicated by L. Erdös

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Wilde, M.M., Winter, A. & Yang, D. Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy. Commun. Math. Phys. 331, 593–622 (2014). https://doi.org/10.1007/s00220-014-2122-x

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