Quantum logarithmic sobolev inequalities for quantum many-body systemsAn approach via quasi-factorization of the relative entropy

Resumen

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi- factorization results for the entropy. Inspired by the classical case, in this thesis we present a strategy to derive the positivity of the logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian, via results of quasi-factorization of the relative entropy. In particular, we address this problem for the heat-bath and Davies dynamics. In the first part of the thesis, we introduce the notion of conditional relative entropy in several ways, and subsequently use these concepts to obtain results of quasi- factorization of the relative entropy both in a weak and strong regime. Next, we show the positivity of logarithmic Sobolev constants for the heat-bath dynamics with tensor product fixed point, and then lift these results for the heat-bath dynamics in 1D and the Davies dynamics, showing that the first one is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi- factorization of the relative entropy, and the second one under some strong clustering of correlations. To conclude, in the last part of the thesis we study a notion related to the conditional relative entropy, namely the Belavkin-Staszewski relative entropy, for which we provide new conditions for equality in the data processing inequality, which we subsequently employ to strengthen the aforementioned inequality for the BS relative entropy in particular, and for maximal f-divergences in general. This thesis has been developped at Instituto de Ciencias Matemáticas and Universidad Autónoma de Madrid under the supervision of David Pérez-García (U. Complutense de Madrid) and Angelo Lucia (Caltech).