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

• Published in: Journal of functional analysis Vol. 273; no. 5; pp. 1810 - 1869
• Main Authors: Carlen, Eric A., Maas, Jan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2017
• Subjects: Detailed balance; Entropy; Gradient flow; Quantum Markov semigroup; 47C90; Detailed balance; Quantum Markov semigroup; 81S22; 46L57; Entropy; Gradient flow; 34D05
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2017.05.003

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.