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

We define the quantum $p$-divergences 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 the quantum Beckner's inequalities interpolate between the Sobolev-type inequalities and the Poincar\'{e} inequality in a sharp way. We provide a uniform lower bound for the Beckner constant $\alpha_p$ in terms of the spectral gap and establish the stability of $\alpha_p$ 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 $\dot{H}^{-1}$ Sobolev distance. We show that the quantum Markov semigroup with $\sigma$-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 Poincar\'{e} inequalities can follow.