Quantum conditional relative entropy and quasi-factorization of the relative entropy
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-σ-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.- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- November 2018
- DOI:
- 10.1088/1751-8121/aae4cf
- arXiv:
- arXiv:1804.09525
- Bibcode:
- 2018JPhA...51V4001C
- Keywords:
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- Quantum Physics;
- Computer Science - Information Theory;
- Mathematical Physics
- E-Print:
- v2: Final version, minor changes, 39 pages, 2 figures