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

E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869.


Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2017-09-01
Journal Title
Journal of Functional Analysis
Volume
273
Issue
5
Page
1810 - 1869
ISSN
IST-REx-ID

Cite this

Carlen E, Maas J. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis. 2017;273(5):1810-1869. doi:10.1016/j.jfa.2017.05.003
Carlen, E., & Maas, J. (2017). Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis, 273(5), 1810–1869. https://doi.org/10.1016/j.jfa.2017.05.003
Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis 273, no. 5 (2017): 1810–69. https://doi.org/10.1016/j.jfa.2017.05.003.
E. Carlen and J. Maas, “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance,” Journal of Functional Analysis, vol. 273, no. 5, pp. 1810–1869, 2017.
Carlen E, Maas J. 2017. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis. 273(5), 1810–1869.
Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis, vol. 273, no. 5, Academic Press, 2017, pp. 1810–69, doi:10.1016/j.jfa.2017.05.003.

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