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Paper   IPM / M / 16402
School of Mathematics
  Title:   Quantum reverse hypercontractivity: its tensorization and application to strong converses
  Author(s):  Salman Beigi (Joint with N. Datta and C. Rouze)
  Status:   Published
  Journal: Commun. Math. Phys.
  Vol.:  376
  Year:  2020
  Pages:   753-794
  Supported by:  IPM
  Abstract:
In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock-Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

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