@article{956,
abstract = {We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. 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 σ 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.},
author = {Carlen, Eric and Maas, Jan},
issn = {00221236},
journal = {Journal of Functional Analysis},
number = {5},
pages = {1810 -- 1869},
publisher = {Academic Press},
title = {{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance}},
doi = {10.1016/j.jfa.2017.05.003},
volume = {273},
year = {2017},
}