Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
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DOI：
10.1016/j.jfa.2017.05.003
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发表时间：
2017-09-01
影响因子：
1.7
通讯作者：
Maas, Jan
中科院分区：
文献类型：
--
作者：
Carlen, Eric A.;Maas, Jan
研究了有限维单位C*-代数上的一类遍历量子马氏半群。这些半群有一个唯一的稳态σ，我们关注的是那些满足量子详细平衡条件的σ。我们表明，这样一个量子马尔可夫半群的状态集的演化是梯度流的相对熵相对于西格玛在一个特定的黎曼度量的状态集。这个度量是2-Wasserstein度量的一个非交换的类似物，在几个有趣的情况下，我们能够证明，与奥托关于经典2-Wasserstein度量的梯度流的工作类似，相对熵相对于这里介绍的黎曼度量是严格一致凸的。由此得到了详细平衡遍历量子马氏半群的相对熵衰减的若干新的不等式。(C)2017爱思唯尔公司版权所有© 2016
We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C*-algebras. These semigroups have a unique stationary state sigma, and we are concerned with those that satisfy a quantum detailed balance condition with respect to sigma. 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 sigma 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. (C) 2017 Elsevier Inc. All rights reserved..