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.

中文翻译：

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量子逆超收缩性：其张量化及其在强变换中的应用

在本文中，我们发展了量子反向超收缩不等式的理论，并展示了如何从 log-Sobolev 不等式中推导出它们。接下来，我们证明了非交换设置中 Stroock-Varopoulos 不等式的推广，这使我们能够分别从 2-log-Sobolev 和 1-log-Sobolev 不等式中导出量子超收缩性和反向超收缩性不等式。然后，我们证明了一些张量化类型的结果，为我们提供了证明超收缩性和反向超收缩性的工具，不仅适用于某些量子超级算子，还适用于它们的张量功率。最后，作为这些结果的应用，我们概括了最近的技术，该技术通过对量子设置的反向超收缩不等式来证明信息论中的强逆界。