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The minimum Rényi entropy output of a quantum channel is locally additive

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Abstract

We show that the minimum Rényi entropy output of a quantum channel is locally additive for Rényi parameter \(\alpha >1\). While our work extends the results of Gour and Friedland (IEEE Trans. Inf. Theory 59(1):603, 2012) (in which local additivity was proven for \(\alpha =1\)), it is based on several new techniques that incorporate the multiplicative nature of \(\ell _p\)-norms, in contrast to the additivity property of the von-Neumann entropy. Our results demonstrate that the counterexamples to the Rényi additivity conjectures exhibit purely global effects of quantum channels. Interestingly, the approach presented here cannot be extended to Rényi entropies with parameter \(\alpha <1\).

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Acknowledgements

We extend thanks to Mark Girard for many stimulating discussions on topics that are closely related to this work.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

    Gilad Gour

  2. Department of Mathematics, University of California, 9500 Gilman Drive, La Jolla, San Diego, CA, 92093-0112, USA

    Todd Kemp

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  1. Gilad Gour
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  2. Todd Kemp
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Correspondence to Gilad Gour.

Additional information

G. Gour was supported by NSERC; T. Kemp was supported by NSF CAREER Award DMS-1254807.

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Gour, G., Kemp, T. The minimum Rényi entropy output of a quantum channel is locally additive. Lett Math Phys 107, 1131–1155 (2017). https://doi.org/10.1007/s11005-016-0933-8

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