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Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

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Abstract

For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.

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Authors and Affiliations

  1. School of Computer Science, McGill University, Montreal, Quebec, H3A 2A7, Canada

    Patrick Hayden

  2. Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Andreas Winter

  3. Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore, 117542, Singapore

    Andreas Winter

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  1. Patrick Hayden
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  2. Andreas Winter
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Correspondence to Andreas Winter.

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Communicated by M.B. Ruskai

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Hayden, P., Winter, A. Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1. Commun. Math. Phys. 284, 263–280 (2008). https://doi.org/10.1007/s00220-008-0624-0

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