  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.
