Interpolation Between Modified Logarithmic Sobolev and Poincaré Inequalities for Quantum Markovian Dynamics

We define the quantum p-divergence and introduce Beckner’s inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative L_p -norm. We obtain a number of implications between Beckner’s inequalities and other quantum functional inequalities, as well as the hypercontractivity. In particular, we show that quantum Beckner’s inequalities interpolate between Sobolev-type inequalities and Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constant α_p in terms of the spectral gap and establish the stability of α_p with respect to the invariant state. As applications, we compute the Beckner constant for the depolarizing semigroup and discuss the mixing time. For symmetric quantum Markov semigroups, we derive the moment estimate, 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, 2017) and a noncommutative H˙ ^-1 Sobolev distance. We show that the quantum Markov semigroup with σ -GNS detailed balance is the gradient flow of a 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 Poincaré inequalities can follow. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.