Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

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 hypercontractivi...

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Bibliographic Details
Published in:Communications in mathematical physics 2020-06, Vol.376 (2), p.753-794
Main Authors: Beigi, Salman, Datta, Nilanjana, Rouzé, Cambyse
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Hypothesis testing
Inequalities
Information theory
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Tensors
Theoretical
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Summary: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03750-z