Cohomological Aspects of Entanglement Entropy: From Information Theory to Noncommutative Geometry
Abstract
We develop a cohomological framework for entanglement entropy that unifies information theory, operator algebras, and noncommutative geometry. Entropy is a 1-cocycle in information cohomology; we generalize this to the quantum setting via Hochschild and cyclic cohomology. For an inclusion of von Neumann algebras N contained in M with a faithful normal state w admitting a cyclic and separating vector, and a conditional expectation E: M -> N (whose existence is equivalent to modular invariance of N by Takesaki's theorem), we construct the entanglement complex as the image of the transgression map tau_w on the relative cyclic complex, where A is a dense *-subalgebra of M and B = A intersect N. Our main result identifies entanglement cohomology with the kernel of the restriction map HC*(A) -> HC*(B), up to the connecting homomorphism d of the long exact sequence of the pair (A,B); when d = 0, the identification is exact. The relative modular cocycle u_t = (Dw : D(w|_N) o E)_t generates explicit entanglement classes when E is not w-invariant, with the Araki relative entropy as its infinitesimal generator. The framework accommodates Type III von Neumann algebras, where no local density matrix exists, and leads to a holographic proposal: a subspace of H^1_ent is conjecturally dual to linearized modular Hamiltonian perturbations of the bulk entanglement wedge. We verify the construction in finite-dimensional, type II_1, and type III_1 examples.
Related
BibTeX
@misc{rashkov2026cohomological,
title = {{Cohomological Aspects of Entanglement Entropy: From Information Theory to Noncommutative Geometry}},
author = {R. C. Rashkov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17195},
eprint = {2609.17195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17195},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cohomological Aspects of Entanglement Entropy: From Information Theory to Noncommutative Geometry
AU - R. C. Rashkov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.17195
UR - https://arxiv.org/abs/2609.17195
AB - We develop a cohomological framework for entanglement entropy that unifies information theory, operator algebras, and noncommutative geometry. Entropy is a 1-cocycle in information cohomology; we generalize this to the quantum setting via Hochschild and cyclic cohomology. For an inclusion of von Neumann algebras N contained in M with a faithful normal state w admitting a cyclic and separating vector, and a conditional expectation E: M -> N (whose existence is equivalent to modular invariance of N by Takesaki's theorem), we construct the entanglement complex as the image of the transgression map tau_w on the relative cyclic complex, where A is a dense *-subalgebra of M and B = A intersect N. Our main result identifies entanglement cohomology with the kernel of the restriction map HC*(A) -> HC*(B), up to the connecting homomorphism d of the long exact sequence of the pair (A,B); when d = 0, the identification is exact. The relative modular cocycle u_t = (Dw : D(w|_N) o E)_t generates explicit entanglement classes when E is not w-invariant, with the Araki relative entropy as its infinitesimal generator. The framework accommodates Type III von Neumann algebras, where no local density matrix exists, and leads to a holographic proposal: a subspace of H^1_ent is conjecturally dual to linearized modular Hamiltonian perturbations of the bulk entanglement wedge. We verify the construction in finite-dimensional, type II_1, and type III_1 examples.
ER -