The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $e^{O(\log^D(n/\epsilon))}$, which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.

翻译：马尔可夫性质蕴含了一般经典哈密顿量的吉布斯分布所固有的条件独立结构，这一特性在推断、混合时间分析和算法设计中起着至关重要的作用。然而，对于量子吉布斯态，相关认知则少得多。本工作中，我们证明对于任何具有有限相互作用度的哈密顿量，量子吉布斯态在任意温度下都是局部马尔可夫的，这意味着对于每个局部区域都存在一个准局域的恢复映射。值得注意的是，该恢复映射是通过施加一个在区域上跃迁的细致平衡Lindbladian而获得的。由此，我们证明了：(i) 对于一个被屏蔽的小区域，其条件互信息（CMI）随屏蔽距离呈指数衰减；(ii) 在关联一致聚类的假设下，$D$维晶格上一般非对易哈密顿量的吉布斯态可以通过深度为$e^{O(\log^D(n/\epsilon))}$的量子电路制备，若假设特定的局部能隙条件，该深度还可进一步降低。我们的证明引入了一种在任意低温下对虚时演化算符的正则化方案，并揭示了Dirichlet形式（一种动力学量）与KMS内积中的对易子（一种静态量）之间的联系。我们相信这些工具为应对量子热力学和混合时间（尤其是在低温区域）中的进一步挑战铺平了道路。