We show that for weakly dependent random variables the relative entropy functional satisfies an approximate version of the stan- dard tensorization property which holds in the independent case. As a corollary we obtain a family of dimensionless logarithmic Sobolev inequal- ities. In the context of spin systems on a graph, the weak dependence requirements resemble the well known Dobrushin uniqueness conditions. Our results can be considered as a discrete counterpart of a recent work of Katalin Marton . We also discuss some natural generalizations such as approximate Shearer estimates and subadditivity of entropy.
Caputo, P., Menz, G., & Tetali, P. (2015). Approximate tensorization of entropy at high temperature. ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE., 24(4), 691-716.