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Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance | | Eric A. Carlen
; Jan Maas
; | | Date: |
5 Sep 2016 | | Abstract: | We study a class of ergodic quantum Markov semigroups on finite-dimensional
unital $C^*$-algebras. These semigroups have a unique stationary state
$sigma$, and we are concerned with those that satisfy a quantum detailed
balance condition with respect to $sigma$. 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 $sigma$ 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. | | Source: | arXiv, 1609.1254 | | Services: | Forum | Review | PDF | Favorites | |
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