标题： 条件互信息的聚类与任意温度下的量子马尔可夫结构 显示英文标题

标题： Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures

摘要： 最近的研究揭示了奇特的量子相，这些相无法通过简单的双部分关联函数进行表征。 在这些相中，来自三部分关联的长程纠缠起着核心作用。 因此，研究多部分关联已成为现代物理学的焦点。 在这些研究中，条件互信息（CMI）是最为确立的信息论度量之一，擅长捕捉各种奇特相的本质，包括拓扑有序的相。 在量子多体物理领域，建立一个量子版的Hammersley-Clifford定理一直是长期追求的目标，该定理连接了吉布斯态和马尔可夫网络这两个概念。 该定理指出，在所有热平衡量子相中，CMI的相关长度保持短程。 在本工作中，我们证明CMI相对于距离呈指数衰减，其相关长度随逆温度的多项式增长。 虽然这一聚类定理之前已在无热相变的高温下得到确立，但在低温下仍难以捉摸，因为在低温下真实的长程纠缠已被量子拓扑序所证实。 我们的发现表明，即使在低温下，一大类三部分纠缠也无法在长程区域表现出来。 为了实现这一证明，我们建立了一个全面的形式化方法来分析子系统上有效哈密顿量的局域性，通常称为“纠缠哈密顿量”或“平均力哈密顿量”。 作为我们分析的一个结果，我们改进了先前关于双部分纠缠的聚类定理。 本质上，这意味着我们研究的是超越正部分转置（PPT）类限制的真实双部分纠缠。 显示英文摘要

摘要： Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. In these, Conditional Mutual Information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay concerning distance, with its correlation length increasing polynomially with the inverse temperature. While this clustering theorem has previously been established for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the `entanglement Hamiltonian' or `Hamiltonian of mean force.' As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, this means that we investigate genuine bipartite entanglement that extends beyond the limitations of the Positive Partial Transpose (PPT) class.