论文标题
相对熵衰减和完整的阳性混合时间
Relative entropy decay and complete positivity mixing time
论文作者
论文摘要
我们证明，量子马尔可夫半群的完整修改对数Sobolev常数受其完整阳性混合时间的逆。对于经典的马尔可夫半群，这意味着hörmander系统在紧凑的歧管上给出的每个亚拉皮拉良好都满足了矩阵值值函数的均匀修饰的log-sobolev不平等。对于量子马尔可夫半群，我们获得了完整的修改对数Sobolev常数与光谱间隙相当，达到尺寸常数的对数的常数。对于量子出生死亡过程，该估计值在渐近上很紧。我们的结果和浓度不平等的结果适用于冯·诺伊曼代数的GNS对称半群。
We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this implies that every sub-Laplacian given by a Hörmander system on a compact manifold satisfies a uniform modified log-Sobolev inequality for matrix-valued functions. For quantum Markov semigroups, we obtain that the complete modified logarithmic Sobolev constant is comparable to spectral gap up to a constant as logarithm of dimension constant. This estimate is asymptotically tight for a quantum birth-death process. Our results and the consequence of concentration inequalities apply to GNS-symmetric semigroups on general von Neumann algebras.
