---
_id: '956'
abstract:
- lang: eng
text: 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.
article_processing_charge: No
author:
- first_name: Eric
full_name: Carlen, Eric
last_name: Carlen
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: 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
apa: Carlen, E., & Maas, J. (2017). Gradient flow and entropy inequalities for
quantum Markov semigroups with detailed balance. Journal of Functional Analysis.
Academic Press. https://doi.org/10.1016/j.jfa.2017.05.003
chicago: Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for
Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis.
Academic Press, 2017. https://doi.org/10.1016/j.jfa.2017.05.003.
ieee: 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. Academic Press, pp. 1810–1869, 2017.
ista: 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.
mla: 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.
short: E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869.
date_created: 2018-12-11T11:49:24Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2023-09-22T10:00:18Z
day: '01'
department:
- _id: JaMa
doi: 10.1016/j.jfa.2017.05.003
external_id:
isi:
- '000406082300005'
intvolume: ' 273'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1609.01254
month: '09'
oa: 1
oa_version: Submitted Version
page: 1810 - 1869
publication: Journal of Functional Analysis
publication_identifier:
issn:
- '00221236'
publication_status: published
publisher: Academic Press
publist_id: '6452'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed
balance
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 273
year: '2017'
...