Additivity rates and PPT property for random quantum channels
Annales mathématiques Blaise Pascal, Volume 22 (2015) no. 1, pp. 1-72.

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 $p\ge 30.95$.

DOI: 10.5802/ambp.345
Classification: 46L54,  60B20,  81P45
Keywords: Random matrix, Free Probability, Quantum Channel, Entropy, Additivity
Motohisa Fukuda 1; Ion Nechita 2, 1

1 Zentrum Mathematik, M5 Technische Universität München Boltzmannstrasse 3 85748 Garching (Germany)
2 CNRS, Laboratoire de Physique Théorique, IRSAMC Université de Toulouse, UPS F-31062 Toulouse (France)
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Motohisa Fukuda; Ion Nechita. Additivity rates and PPT property for random quantum channels. Annales mathématiques Blaise Pascal, Volume 22 (2015) no. 1, pp. 1-72. doi : 10.5802/ambp.345. https://ambp.centre-mersenne.org/articles/10.5802/ambp.345/

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