### 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 language | English (US) |
---|---|

Pages (from-to) | 1810-1869 |

Number of pages | 60 |

Journal | Journal of Functional Analysis |

Volume | 273 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2017 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis

### Keywords

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

### Cite this

*Journal of Functional Analysis*,

*273*(5), 1810-1869. https://doi.org/10.1016/j.jfa.2017.05.003

}

*Journal of Functional Analysis*, vol. 273, no. 5, pp. 1810-1869. https://doi.org/10.1016/j.jfa.2017.05.003

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

Research output: Contribution to journal › Article

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 -