Logarithmic Sobolev Inequalities for Quantum Many-Body Systems.

          @misc{ scivideos_PIRSA:19100062,
            doi = {10.48660/19100062},
            url = {https://pirsa.org/19100062},
            author = {Capel, Angela},
            keywords = {Other Physics},
            language = {en},
            title = {Logarithmic Sobolev Inequalities for Quantum Many-Body Systems.},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2019},
            month = {oct},
            note = {PIRSA:19100062 see, \url{https://scivideos.org/pirsa/19100062}}
          }
          

Abstract

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, 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. In particular we address this problem 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.