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Blaise Pascal
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Motohisa Fukuda; Ion Nechita
Additivity rates and PPT property for random quantum channels
Annales mathématiques Blaise Pascal, 22 no. 1 (2015), p. 1-72, doi: 10.5802/ambp.345
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Class. Math.: 46L54, 60B20, 81P45
Keywords: Random matrix, Free Probability, Quantum Channel, Entropy, Additivity

Résumé - Abstract

Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel $L$, which give the first order (linear) term of the minimum output $p$-Rényi entropies of $L^{\otimes r}$ as functions of $r$. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the $p$-Rényi entropy for all $p\ge 30.95$.

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