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

Eric Carlen, Jan Maas

Research output: Contribution to journalArticle

13 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)1810-1869
Number of pages60
JournalJournal of Functional Analysis
Volume273
Issue number5
DOIs
StatePublished - Sep 1 2017

Fingerprint

Markov Semigroups
Entropy Inequality
Detailed Balance
Gradient Flow
Relative Entropy
Wasserstein Metric
Riemannian Metric
Uniformly Convex
Strictly Convex
Stationary States
Unital
C*-algebra
Analogy
Semigroup
Decay
Analogue
Metric

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Detailed balance
  • Entropy
  • Gradient flow
  • Quantum Markov semigroup

Cite this

@article{071aecd0cfef43a1bfb3f6ed4bf51a43,
title = "Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance",
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.",
keywords = "Detailed balance, Entropy, Gradient flow, Quantum Markov semigroup",
author = "Eric Carlen and Jan Maas",
year = "2017",
month = "9",
day = "1",
doi = "10.1016/j.jfa.2017.05.003",
language = "English (US)",
volume = "273",
pages = "1810--1869",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "5",

}

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. / Carlen, Eric; Maas, Jan.

In: Journal of Functional Analysis, Vol. 273, No. 5, 01.09.2017, p. 1810-1869.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

AU - Carlen, Eric

AU - Maas, Jan

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

KW - Detailed balance

KW - Entropy

KW - Gradient flow

KW - Quantum Markov semigroup

UR - http://www.scopus.com/inward/record.url?scp=85020287680&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020287680&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2017.05.003

DO - 10.1016/j.jfa.2017.05.003

M3 - Article

AN - SCOPUS:85020287680

VL - 273

SP - 1810

EP - 1869

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 5

ER -