The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated with the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular, we show that for the heat-bath dynamics of 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

中文翻译：

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一维系统热浴动力学的修正对数 Sobolev 不等式

开放量子多体系统的马尔可夫耗散演化的混合时间可以使用某些量子函数不等式的最优常数来限制，例如修正的对数 Sobolev 常数。对于经典自旋系统，此类常数的正值来自于 Gibbs 测度的混合条件，通过熵的准因式分解结果。受经典案例的启发，我们提出了一种策略，从局部交换哈密顿量的 Gibbs 状态的一些聚类条件导出与某些量子系统的动力学相关的修正对数 Sobolev 常数的正值。特别是，我们表明对于一维系统的热浴动力学，