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

By Eric A. Carlen, Jan Maas. 2016

212 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7aec5b6a859ec64aa1969365d0ae208f74e75c2