Quantum conditional relative entropy and quasifactorization of the relative entropy
Abstract
The existence of a positive logSobolev 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 quasifactorization of the entropy in terms of the conditional entropy in some subσalgebras.
In this work we analyze analogous quasifactorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasifactorization results for it. As an illustration of their potential, we use one of them to obtain a positive logSobolev constant for the heatbath dynamics with product fixed point.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 November 2018
 DOI:
 10.1088/17518121/aae4cf
 arXiv:
 arXiv:1804.09525
 Bibcode:
 2018JPhA...51V4001C
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory;
 Mathematical Physics
 EPrint:
 v2: Final version, minor changes, 39 pages, 2 figures