  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 modified 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 modified 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 show that for
the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev
constant is positive under the assumptions of a mixing condition on the Gibbs
state and a strong quasi-factorization of the relative entropy.
