Quasi-factorization and multiplicative comparison of subalgebra-relative entropy

Purely multiplicative comparisons of quantum relative entropy are desirable but challenging to prove. We show such for entropies between comparable densities, including the a density with respect its subalgebraic restriction. These inequalities asymptotically tight in approaching known, as perturbation size approaches zero. Based on these results, we obtain kind inequality known quasi-factorization or approximate tensorization entropy. Quasi-factorization lower bounds sum density's several restrictions terms their intersection's As applications, implies uncertainty-like relations, and an iteration trick, it yields decay estimates optimal asymptotic order mixing processes described by finite, connected, undirected graphs.