### 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) |
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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 |

### All Science Journal Classification (ASJC) codes

- Analysis

### Keywords

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

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## Cite this

*Journal of Functional Analysis*,

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