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

Eric A. Carlen, Jan Maas

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1810-1869
Number of pages60
JournalJournal of Functional Analysis
Issue number5
StatePublished - Sep 1 2017

All Science Journal Classification (ASJC) codes

  • Analysis


  • Detailed balance
  • Entropy
  • Gradient flow
  • Quantum Markov semigroup

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