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Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0

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Abstract

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples.

An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≲ 0.11. We conjecture however that violations of additivity exist for all p < 1.

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Authors and Affiliations

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

    Toby Cubitt & Andreas Winter

  2. Department of Computer Science, University of Bristol, Bristol, BS8 1UB, UK

    Aram W. Harrow & Ashley Montanaro

  3. Institute for Quantum Computing, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Debbie Leung

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

    Andreas Winter

Authors
  1. Toby Cubitt
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  2. Aram W. Harrow
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  3. Debbie Leung
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  4. Ashley Montanaro
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  5. Andreas Winter
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Correspondence to Andreas Winter.

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Communicated by M.B. Ruskai

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Cubitt, T., Harrow, A.W., Leung, D. et al. Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0. Commun. Math. Phys. 284, 281–290 (2008). https://doi.org/10.1007/s00220-008-0625-z

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  • DOI: https://doi.org/10.1007/s00220-008-0625-z

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