Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
Carlen, Eric
Maas, Jan
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.
Academic Press
2017
info:eu-repo/semantics/article
doc-type:article
text
https://research-explorer.app.ist.ac.at/record/956
Carlen E, Maas J. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. <i>Journal of Functional Analysis</i>. 2017;273(5):1810-1869. doi:<a href="https://doi.org/10.1016/j.jfa.2017.05.003">10.1016/j.jfa.2017.05.003</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jfa.2017.05.003
info:eu-repo/semantics/altIdentifier/issn/00221236
info:eu-repo/semantics/openAccess