Quantum Reverse Hypercontractivity for KMS-Symmetric Quantum Markov Semigroups

Abstract

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 $n \geq 3$.