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We define the quantum $p$-divergences and introduce Beckner's inequalitiesfor primitive quantum Markov semigroups on a finite-dimensional matrix algebrasatisfying the detailed balance condition. Such inequalities quantify theconvergence rate of the quantum dynamics in the noncommutative $L_p$-norm. Weobtain a number of implications between Beckner's inequalities and otherquantum functional inequalities, as well as the hypercontractivity. Inparticular, we show that the quantum Beckner's inequalities interpolate betweenthe 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 termsof the spectral gap and establish the stability of $\alpha_p$ with respect tothe invariant state. As applications, we compute the Beckner constant for thedepolarizing semigroup and discuss the mixing time. For symmetric quantumMarkov semigroups, we derive the moment estimate, which further implies aconcentration 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 $\dot{H}^{-1}$ Sobolevdistance. We show that the quantum Markov semigroup with $\sigma$-GNS detailedbalance is the gradient flow of a quantum $p$-divergence with respect to themetric $W_{2,p}$. We prove that the set of quantum states equipped with$W_{2,p}$ is a complete geodesic space. We then consider the associatedentropic Ricci curvature lower bound via the geodesic convexity of$p$-divergence, and obtain an HWI-type interpolation inequality. This enablesus to prove that the positive Ricci curvature implies the quantum Beckner'sinequality, from which a transport cost and Poincar\'{e} inequalities canfollow.

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