Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Abstract

In this work, we define the quantum p-divergences and introduce the Beckner's inequalities for the primitive quantum Markov semigroups with σ-detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative Lp-norm. We obtain a number of implications between Beckner's inequalities and other quantum functional inequalities. In particular, we show that the quantum Beckner's inequalities interpolate between the Sobolev-type inequalities and the Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constants αp in terms of the spectral gap, and establish the stability of αp with respect to the invariant state. As applications of Beckner's inequalities, we obtain an improved bound for the mixing time. For the symmetric quantum Markov semigroups, we derive moment estimates, which further implies a concentration inequality.We introduce a new class of quantum transport distances W 2,p interpolating the quantum 2-Wasserstein distance by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869] and a noncommutative Ḣ−1 Sobolev distance. We show that the quantum Markov semigroup with σ-detailed balance is the gradient flow of the quantum p-divergence with respect to the metric W 2,p . We prove that the set of quantum states equipped with W 2,p is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of p-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner's inequality, from which a transport cost and the Poincaré inequalities can follow.