纠缠熵的上同调方面：从信息论到非交换几何
Cohomological Aspects of Entanglement Entropy: From Information Theory to Noncommutative Geometry
- Sofia University（索非亚大学）
- Vienna University of Technology（维也纳工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出统一信息论、算子代数与非交换几何的纠缠熵上同调框架，通过条件期望嵌入Connes循环双复形，以模理论刻画相对纠缠，为无局部密度矩阵的III型代数及全息纠缠提供严格基础。
AI中文摘要：
我们发展了一个纠缠熵的上同调框架，该框架统一了信息论、算子代数和非交换几何的视角。从熵作为1-上圈的信息论刻画出发，我们展示了这一结构如何通过Hochschild和循环上同调推广到量子环境。一个核心结果是通过条件期望将纠缠复形嵌入Connes循环双复形，从而将纠缠上同调识别为从von Neumann代数到其子代数的限制映射的核。Tomita-Takesaki模理论提供了动力学结构，其中Connes-Radon-Nikodym上圈作为编码相对纠缠的基本对象。该框架自然容纳了不存在局部密度矩阵的III型von Neumann代数，为量子场论和全息中的纠缠提供了严格基础。
英文摘要：
We develop a cohomological framework for entanglement entropy that unifies perspectives from information theory, operator algebras, and noncommutative geometry. Starting from the information-theoretic characterization of entropy as a 1-cocycle, we show how this structure generalizes to the quantum setting through Hochschild and cyclic cohomology. A central result is the embedding of an entanglement complex into the Connes cyclic bicomplex via a conditional expectation, identifying entanglement cohomology as the kernel of the restriction map from a von Neumann algebra to its subalgebra. The Tomita-Takesaki modular theory provides the dynamical structure, with the Connes-Radon-Nikodym cocycle serving as the fundamental object encoding relative entanglement. This framework naturally accommodates Type III von Neumann algebras, where no local density matrix exists, offering a rigorous foundation for entanglement in quantum field theory and holography.
