  Given a uniform, frustration-free family of local Lindbladians defined on a
quantum lattice spin system in any spatial dimension, we prove a strong
exponential convergence in relative entropy of the system to equilibrium under
a condition of spatial mixing of the stationary Gibbs states and the rapid
decay of the relative entropy on finite-size blocks. Our result leads to the
first examples of the positivity of the modified logarithmic Sobolev inequality
for quantum lattice spin systems independently of the system size. Moreover, we
show that our notion of spatial mixing is a consequence of the recent quantum
generalization of Dobrushin and Shlosman's complete analyticity of the
free-energy at equilibrium. The latter typically holds above a critical
temperature Tc. Our results have wide-ranging applications in quantum
information. As an illustration, we discuss four of them: first, using
techniques of quantum optimal transport, we show that a quantum annealer
subject to a finite range classical noise will output an energy close to that
of the fixed point after constant annealing time. Second, we prove Gaussian
concentration inequalities for Lipschitz observables and show that the
eigenstate thermalization hypothesis holds for certain high-temperture Gibbs
states. Third, we prove a finite blocklength refinement of the quantum Stein
lemma for the task of asymmetric discrimination of two Gibbs states of
commuting Hamiltonians satisfying our conditions. Fourth, in the same setting,
our results imply the existence of a local quantum circuit of logarithmic depth
to prepare Gibbs states of a class of commuting Hamiltonians.
