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Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication

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Abstract

This paper studies the difficulty of discriminating between an arbitrary quantum channel and a “replacer" channel that discards its input and replaces it with a fixed state. The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein’s lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel \({\mathcal{N}}\) as the optimal Type II error exponent when discriminating between a large number of independent instances of \({\mathcal{N}}\) and an arbitrary “worst-case” replacer channel chosen from the set of all replacer channels.

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Authors and Affiliations

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

    Tom Cooney & Mark M. Wilde

  2. Física Teòrica: Informació i Fenòmens Quàntics, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Milán Mosonyi

  3. Mathematical Institute, Budapest University of Technology and Economics, Egry József u 1., Budapest, 1111, Hungary

    Milán Mosonyi

  4. Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Mark M. Wilde

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  1. Tom Cooney
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  2. Milán Mosonyi
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Correspondence to Mark M. Wilde.

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Communicated by M. M. Wolf

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Cooney, T., Mosonyi, M. & Wilde, M.M. Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication. Commun. Math. Phys. 344, 797–829 (2016). https://doi.org/10.1007/s00220-016-2645-4

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  • DOI: https://doi.org/10.1007/s00220-016-2645-4

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