Additivity rates and PPT property for random quantum channels
Annales Mathématiques Blaise Pascal, Tome 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 : https://doi.org/10.5802/ambp.345
Classification : 46L54,  60B20,  81P45
Mots clés : Random matrix, Free Probability, Quantum Channel, Entropy, Additivity
@article{AMBP_2015__22_1_1_0,
author = {Fukuda, Motohisa and Nechita, Ion},
title = {Additivity rates and PPT property for random quantum channels},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {1--72},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {22},
number = {1},
year = {2015},
doi = {10.5802/ambp.345},
zbl = {1338.46072},
mrnumber = {3361563},
language = {en},
url = {www.numdam.org/item/AMBP_2015__22_1_1_0/}
}
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. http://www.numdam.org/item/AMBP_2015__22_1_1_0/

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