Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy

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.

55 pages, 2 figures; corrections and clarifications

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Cited by in corpus (6)

• Rapid thermalization of spin chain commuting Hamiltonians
• Complete entropic inequalities for quantum Markov chains
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Approximate tensorization of the relative entropy for noncommuting conditional expectations
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Approximate Unitary $k$-Designs from Shallow, Low-Communication Circuits