Additivity rates and PPT property for random quantum channels

Fukuda, Motohisa; Nechita, Ion

Annales mathématiques Blaise Pascal, Tome 22 (2015), p. 1-72 / Harvested from Numdam

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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 $p \geq 30.95$ .

Publié le : 2015-01-01

MR 3361563 | Zbl 1338.46072

DOI : https://doi.org/10.5802/ambp.345
Classification: 46L54, 60B20, 81P45

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     author = {Fukuda, Motohisa and Nechita, Ion},
     title = {Additivity rates and PPT property for random quantum channels},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {22},
     year = {2015},
     pages = {1-72},
     doi = {10.5802/ambp.345},
     mrnumber = {3361563},
     zbl = {1338.46072},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2015__22_1_1_0}
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Fukuda, Motohisa; Nechita, Ion. Additivity rates and PPT property for random quantum channels. Annales mathématiques Blaise Pascal, Tome 22 (2015) pp. 1-72. doi : 10.5802/ambp.345. http://gdmltest.u-ga.fr/item/AMBP_2015__22_1_1_0/

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