On contraction coefficients, partial orders and approximation of capacities for quantum channels

Christoph Hirche1, Cambyse Rouzé2,3, and Daniel Stilck França2,3

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
2Department of Mathematics, Technische Universität München, 85748 Garching, Germany
3Munich Center for Quantum Science and Technology (MCQST), München, Germany

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The data processing inequality is the most basic requirement for any meaningful measure of information. It essentially states that distinguishability measures between states decrease if we apply a quantum channel and is the centerpiece of many results in information theory. Moreover, it justifies the operational interpretation of most entropic quantities. In this work, we revisit the notion of contraction coefficients of quantum channels, which provide sharper and specialized versions of the data processing inequality. A concept closely related to data processing is partial orders on quantum channels. First, we discuss several quantum extensions of the well-known less noisy ordering and relate them to contraction coefficients. We further define approximate versions of the partial orders and show how they can give strengthened and conceptually simple proofs of several results on approximating capacities. Moreover, we investigate the relation to other partial orders in the literature and their properties, particularly with regard to tensorization. We then examine the relation between contraction coefficients with other properties of quantum channels such as hypercontractivity. Next, we extend the framework of contraction coefficients to general f-divergences and prove several structural results. Finally, we consider two important classes of quantum channels, namely Weyl-covariant and bosonic Gaussian channels. For those, we determine new contraction coefficients and relations for various partial orders.

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