On the modified logarithmic Sobolev inequality for the heatbath dynamics for 1D systems
Abstract
The mixing time of Markovian dissipative evolutions of open quantum manybody systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure via quasifactorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated with the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular, we show that for the heatbath dynamics of 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasifactorization of the relative entropy.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 June 2021
 DOI:
 10.1063/1.5142186
 arXiv:
 arXiv:1908.09004
 Bibcode:
 2021JMP....62f1901B
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 28 pages, 4 figures. Included some additional comments and updated results in light of recent advances in the literature