Given a thermal quantum dissipative evolution modelled by a quantum Markov semigroup, its mixing time (i.e. time of convergence to its thermal equilibrium) can be bounded using optimal constants associated to a family of non-commutative functional inequalities.

In this talk, we will focus on the so-called “modified logarithmic Sobolev inequality” (MLSI), which can be interpreted as the exponential decay rate for the relative entropy between an evolved state and the thermal equilibrium. The existence of an optimal constant for such an inequality is a sufficient condition for a quantum spin system to satisfy “rapid mixing”, a property with strong implications in various contexts.

For classical spin systems, the problem of estimating MLSI constants, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, is frequently addressed via a result of quasi-factorization of the entropy in terms of conditional entropies in some sub-algebras.

In the past few years, we have extended such a technique to the quantum realm, where we have provided a strategy to prove quantum MLSIs under some decay of correlations on Gibbs states, via results of quasi-factorization of the quantum relative entropy. In this talk, we will present this strategy to prove quantum MLSIs for quantum Markov semigroups modelling thermal dissipative evolutions. We will finish with some specific examples of application of such a strategy to relevant quantum dynamics, with consequences, in particular, in the contexts of quantum memory devices, Gibbs states preparation and dissipative phases of matter.