发表机构
Tsinghua University; Beijing Institute of Mathematical Sciences and Applications（清华大学; 北京数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限模高斯量子马尔可夫半群，给出完全修正对数Sobolev不等式的最优常数，证明Hurwitz漂移下的超压缩性，并发现$p$-对数Sobolev常数在$0<p<1/2$时可为负。
AI 中文摘要
本文研究了具有忠实不变高斯态的有限模高斯量子马尔可夫半群。我们给出了相对于不动点代数的完全修正对数Sobolev不等式的代数刻画，并在漂移演化$e^{t\mathbf{Z}}$当$t\to\infty$时收敛的假设下，得到了相应的最优常数。我们建立了具有Hurwitz漂移的此类量子马尔可夫半群的超压缩性，即每一对固定的$1<p<q<\infty$在足够大的时间上都能达到。我们发现，通常从时间零开始具有正速率的反向超压缩曲线在一般情况下可能失效，但在Hurwitz假设下，每一对固定的$1/2 < p < q < 1$在足够大的时间上都能达到。我们还系统地研究了$p$-对数Sobolev不等式，并证明了当$0 < p < 1/2$时，$p$-对数Sobolev常数可能为负。
英文摘要
In this paper, we study finite-mode Gaussian quantum Markov semigroups with a faithful invariant Gaussian state. We present an algebraic characterization of the complete modified logarithmic Sobolev inequality relative to the fixed-point algebra, and we obtain the corresponding optimal constant under the assumption that the drift evolution $e^{t\mathbf{Z}}$ converges as $t\to\infty$. We establish hypercontractivity for such quantum Markov semigroups with Hurwitz drift, in the sense that every fixed pair $1<p<q<\infty$ is attained at sufficiently large times. We find that the usual reverse hypercontractive curves with a positive rate from time zero can fail in general, but every fixed pair $1/2 < p < q < 1$ is attained at sufficiently large times under the Hurwitz assumption. We also systematically investigate the $p$-logarithmic Sobolev inequality and show that the $p$-logarithmic Sobolev constant can be negative for $0 < p < 1/2$.
