Search

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. [ABSTRACT FROM AUTHOR]

We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N - 3 . For the purely harmonic chain the order of the rate is in [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N. [ABSTRACT FROM AUTHOR]

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C ∗ -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates. [ABSTRACT FROM AUTHOR]