发表机构
School of Mathematics and Statistics, Wuhan University; Wuhan Institute of Quantum Technology（武汉大学数学与统计学院; 武汉量子技术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入BKM强制性常数，证明其正性决定量子马尔可夫半群的指数相对熵衰减，并建立完全修正对数Sobolev不等式与GNS谱隙正性的等价判据，应用于吉布斯采样器。
AI 中文摘要
我们引入了BKM强制性常数的概念，并证明该常数的正性决定了半群的指数相对熵衰减。具体而言，我们证明了修正对数Sobolev常数与BKM强制性常数等价，其等价关系中的常数依赖于渐近条件期望。我们还发现，对于矩阵放大，BKM强制性常数已在量子比特辅助系统中达到，该常数与生成Lindbladian的GNS谱隙一致。作为推论，我们建立了一个判据：完全修正对数Sobolev不等式成立当且仅当半群的GNS谱隙严格为正。上述所有讨论适用于具有忠实渐近条件期望作为长时间平衡的量子马尔可夫半群，且不假设细致平衡条件。作为应用，我们证明了有限维Chen-Kastoryano-Gilyén吉布斯采样器始终满足完全修正对数Sobolev不等式。
英文摘要
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
Comments33 pages
