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