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Quantum Conditional Relative Entropy and Quasi-Factorization of the relative entropy

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MQIW05 - Beyond I.I.D. in information theory

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—algebras.

In this work we analyze analogous quasi-factorization results in the quantum case. For that, we dene 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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