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arXiv · 1804.09525

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Angela Capel·Angelo Lucia·David Pérez-García
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Abstract

The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$\sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.

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Angela Capel, Angelo Lucia, David Pérez-García. 2018-04-17. Quantum conditional relative entropy and quasi-factorization of the relative entropy. https://doi.org/10.1088/1751-8121%2Faae4cf

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