Communications in Mathematical Physics

, Volume 341, Issue 3, pp 885–909 | Cite as

Almost One Bit Violation for the Additivity of the Minimum Output Entropy

  • Serban T. Belinschi
  • Benoît Collins
  • Ion Nechita
Article

Abstract

In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set Kk,t. We also showed that the set Kk,t is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of \({\ell^p}\) norms on the set Kk,t and prove that the maximum is attained on a vector of shape (a, b, . . . , b) where ab. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any \({\varepsilon > 0}\), it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as \({\log 2 -\varepsilon}\). Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Serban T. Belinschi
    • 1
    • 2
    • 3
  • Benoît Collins
    • 4
    • 5
  • Ion Nechita
    • 6
    • 7
  1. 1.CNRS-Institut de Mathématiques de ToulouseToulouse Cedex 9France
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  3. 3.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  4. 4.Department of MathematicsUniversity of OttawaOttawaCanada
  5. 5.Department of MathematicsKyoto UniversityKyotoJapan
  6. 6.Zentrum Mathematik, M5Technische Universität MünchenGarchingGermany
  7. 7.CNRS, Laboratoire de Physique ThéoriqueToulouseFrance

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