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arXiv · 2209.11684

Relative entropy decay and complete positivity mixing time

Li Gao·Marius Junge·Nicholas LaRacuente·Haojian Li
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Abstract

We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this implies that every sub-Laplacian given by a H\"ormander system on a compact manifold satisfies a uniform modified log-Sobolev inequality for matrix-valued functions. For quantum Markov semigroups, we obtain that the complete modified logarithmic Sobolev constant is comparable to spectral gap up to a constant as logarithm of dimension constant. This estimate is asymptotically tight for a quantum birth-death process. Our results and the consequence of concentration inequalities apply to GNS-symmetric semigroups on general von Neumann algebras.

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Li Gao, Marius Junge, Nicholas LaRacuente, Haojian Li. 2022-09-22. Relative entropy decay and complete positivity mixing time. https://arxiv.org/abs/2209.11684

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