Additivity rates and PPT property for random quantum channels

Fukuda, Motohisa; Nechita, Ion

doi:10.5802/ambp.345

Fukuda, Motohisa ¹ ; Nechita, Ion ^1,²

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)

Annales mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 1-72

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Résumé

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 $p \geq 30.95$ .

MR Zbl

DOI : 10.5802/ambp.345

Classification : 46L54, 60B20, 81P45
Keywords: Random matrix, Free Probability, Quantum Channel, Entropy, Additivity

Affiliations des auteurs :

Fukuda, Motohisa ¹ ; Nechita, Ion ^{1, 2}

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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     doi = {10.5802/ambp.345},
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     language = {en},
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Fukuda, Motohisa; Nechita, Ion. Additivity rates and PPT property for random quantum channels. Annales mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 1-72. doi: 10.5802/ambp.345

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