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Journal of Operator Theory

Volume 73, Issue 2, Spring 2015  pp. 333-360.

On the convergence of output sets of quantum channels

Authors:  Benoit Collins (1) Motohisa Fukuda (2) Ion Nechita (3)
Author institution:(1) Departement de Mathematique et Statistique, Universite d'Ottawa, 585 King Edward, Ottawa, ON, K1N6N5 Canada, and Kyoto University, Department of Mathematics, Japan, and CNRS, Institut Camille Jordan Universite Lyon 1, France
(2) Zentrum Mathematik, M5, Technische Universitaet Muenchen, Boltzmannstrasse 3, 85748 Garching, Germany
(3) CNRS, Laboratoire de Physique Theorique, IRSAMC, Universite de Toulouse, UPS, 31062 Toulouse, France

Summary: We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in \textsc{S.T. Belinschi, B. Collins, I. Nechita}, \textit{Invent. Math.} \textbf{190}(2012), 647--697. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences $\Phi_n$ such that along with any other quantum channel $\Xi$, we have convergence of the output set of $\Phi_n$ and $\Phi_n\otimes \Xi$ simultaneously; the case when $\Xi$ is entanglement breaking is investigated in details.

Keywords: random matrices, quantum information theory, random quantum channel

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