Cohomological Aspects of Entanglement Entropy: From Information Theory to Noncommutative Geometry

Abstract

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.