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arXiv · 1609.01254

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Eric A. Carlen·Jan Maas
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Abstract

We study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $\sigma$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $\sigma$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $\sigma$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical $2$-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

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Eric A. Carlen, Jan Maas. 2016-09-05. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. https://arxiv.org/abs/1609.01254

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