Quantum logarithmic Sobolev inequalities and rapid mixing

TL;DR

This paper introduces quantum logarithmic Sobolev inequalities, establishing their relationship with hypercontractivity and spectral gaps, leading to improved bounds on the mixing times of quantum dynamical semigroups relevant for quantum information theory.

Contribution

It develops a new family of quantum logarithmic Sobolev inequalities and links them to hypercontractivity and spectral gaps, providing sharper convergence bounds for quantum semigroups.

Findings

Derived upper bounds for LS constants in terms of spectral gaps

Established improved mixing time bounds for depolarizing semigroup

Applied inequalities to quantum expanders for faster convergence estimates

Abstract

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative $\bL_p$-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained; including the depolarizing semigroup and quantum expanders.