论文标题
修改的对数Sobolev和量子马尔可力动力学的庞加利不平等现象之间的插值
Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics
论文作者
论文摘要
我们定义了量子$ p $ diverences，并在有限维矩阵代数上引入了贝克纳的原始量子马尔可夫半群的不平等，以满足详细的平衡条件。这种不等式量化了非交通$ l_p $ -norm中量子动力学的收敛速率。我们在贝克纳的不平等和其他量子功能不平等以及超收缩性之间获得了许多含义。特别是，我们表明，量子贝克纳的不平等现象与索波列夫型不平等与庞加莱的不平等之间的插值以一种尖锐的方式。我们在光谱差距方面为贝克纳常数$α_p$提供了均匀的下限，并建立了相对于不变状态的$α_p$的稳定性。作为应用，我们计算去极化半群的贝克纳常数，并讨论混合时间。对于对称量子马尔可夫半群，我们得出了力矩估计，这进一步暗示着浓度不平等。 我们介绍了一个新的量子传输距离$ W_ {2，P} $插值Carlen和Maas的量子2-Wasserstein距离[J.功能。肛门。 273（5），1810-1869（2017）]和非交换性$ \ dot {h}^{ - 1} $ sobolev距离。我们表明，具有$σ$ -gns详细余额的量子马尔可夫半群是量子$ p $ -Divergence相对于度量$ w_ {2，p} $的梯度流。我们证明配备$ W_ {2，P} $的量子状态集是一个完整的地理空间。然后，我们考虑通过$ p $ divergence的测量凸度的相关熵曲率下限，并获得HWI-Type插值不平等。这使我们能够证明积极的RICCI曲率意味着量子贝克纳的不平等，从中可以从中产生运输成本和庞加莱的不平等现象。
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 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 $\dot{H}^{-1}$ 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.
