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


Author(s): Carlen, Eric A; Maas, Jan
Article Title: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
Affiliation IST Austria
Keywords: Entropy; gradient flow; Quantum Markov semigroup; Detailed balance
Journal Title: Journal of Functional Analysis
Volume: 273
Issue 5
ISSN: 0022-1236
Publisher: Elsevier  
Date Published: 2017-09-01
Start Page: 1810
End Page: 1869
Copyright Statement: E.C. was partially supported by NSF grant DMS 1501007, and thanks IST Austria for hospitality during a visit in June 2015. E.C. thanks the Mittag-Leffler Institute for hospitality during the final work on this paper. Both authors thank the Erwin Schrödinger Institute in Vienna for hospitality during a visit in June 2016.
URL:
DOI: 10.1016/j.jfa.2017.05.003
Notes: 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.
Open access: yes (repository)
IST Austria Authors
  1. Jan Maas
    25 Maas