AI 中文总结
本文研究KMS对称量子马尔可夫半群的反向超压缩性，构造了阻碍正一致率的例子，并证明在特定参数区间内所有本原半群满足该性质，且对图情形给出结果。
AI 中文摘要
本文构造了本原KMS对称量子马尔可夫半群，其$p$-对数Sobolev常数在$p=1/2$处消失，以及一个基于图的关联例子，其$p$-对数Sobolev常数在$p\to 0^+$时趋于$-\infty$。这些例子阻碍了$(0,1)$上正的一致反向超压缩率。相比之下，我们证明每个本原KMS对称量子马尔可夫半群在$1/2< p\leq q < 1$时满足量子反向超压缩性，并给出了显式的充分时间。对于每个固定的$\delta>0$，足够强的退相干扰动在$[\delta, 1]$上满足一致正对数Sobolev不等式，从而在$\delta< p\leq q<1$（固定$0<\delta<1/2$）时满足量子反向超压缩性。此外，对于连通图且$n\geq 3$，我们建立了基于图的KMS对称量子马尔可夫半群在$\delta< p\leq q< 1$（固定$0<\delta<1$）时的反向超压缩性。
英文摘要
In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the $p$-logarithmic Sobolev constant vanishes at $p=1/2$ and a correlated graph-based example for which the $p$-logarithmic Sobolev constant tends to $-\infty$ as $p\to 0^+$. These examples obstruct a positive uniform reverse-hypercontractivity rate on $(0,1)$. By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for $1/2< p\leq q < 1$ with explicit sufficient times. For each fixed $δ>0$, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on $[δ, 1]$ and hence quantum reverse hypercontractivity for $δ< p\leq q<1$ with fixed $0<δ<1/2$. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for $δ< p\leq q< 1$ with fixed $0<δ<1$ for connected graphs when $n \geq 3$.
