Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy

TL;DR

This paper establishes quasi-factorization inequalities for quantum relative entropy, providing new bounds that relate the entropy of a density to its subalgebraic restrictions, with applications in uncertainty relations and mixing process decay.

Contribution

It introduces quasi-factorization inequalities for quantum relative entropy, advancing the understanding of multiplicative comparisons and their applications.

Findings

Proved asymptotically tight multiplicative comparisons of quantum relative entropy.

Derived quasi-factorization inequalities relating entropies of densities and subalgebras.

Applied results to uncertainty relations and decay estimates in mixing processes.

Abstract

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.