Dear colleagues, it is our pleasure to invite you to the following seminar Speaker: *Angelo Lucia *(Caltech - Division of Physics, Mathematics and Astronomy) Title: *Mixing time of quantum Gibbs samplers* Time: Wednesday, *April 14*, 5PM (Italian time) The abstract follows below in this message. Zoom Link for the seminar <https://us02web.zoom.us/j/86771523774?pwd=SitEVEhPSGRlaEkxTHZmQk9rTmlLZz09> The seminar is a satellite activity of the PhD course "Introduction to coercive inequalities with applications in analysis and probability theory" (professor Boguslaw Zegarlinski - Imperial College, London) Best regards, Raffaella Carbone Abstract. Gibbs samplers are Markovian semigroups whose evolution converges towards the termal equilibrium of a given Hamiltonian (the Gibbs state). The time that the evolution takes, in the worst case, to converge close to the thermal state is known as the mixing time. Knowing the mixing time of a process is useful for many applications, and it can be estimated with various functional inequalities. In the case of classical spin systems on a lattice, it was shown that it is possible to determine the mixing properties of the semigroup from some "static" clustering condition of the thermal state. This suggests the question of whether the same is true for quantum spin systems. In this talk, I will focus on the specific case of Gibbs states of commuting Hamiltonians, and present some recent result in this direction. I will introduce the notion of quantum conditional relative entropy and show how it can be used to prove quasi-factorization (or approximate tensorization) properties of the quantum relative entropy. I will then show how these can be used to bound log-Sobolev constants for product semigroups with heath-bath generators, and under stronger assumptions in more general situations. -- Raffaella Carbone, PhD Probabilità e Statistica Matematica Dipartimento di Matematica dell'Università degli Studi di Pavia