Quantum Entanglement as a Cohomological Obstruction

TL;DR

This paper presents a novel cohomological framework for understanding quantum entanglement, introducing mathematical tools like sheaf cohomology and a Quantum Entanglement Index to characterize and analyze entangled states.

Contribution

It introduces a cohomological perspective on entanglement, including local entanglement groups, a differential-geometric representation, and the Quantum Entanglement Index, advancing theoretical understanding.

Findings

Cohomological obstruction characterizes entanglement.

Introduction of the Quantum Entanglement Index (QEI).

Numerical implementations for quantum many-body models.

Abstract

We recast quantum entanglement as a cohomological obstruction to reconstructing a global quantum state from locally compatible information. We address this by considering presheaf cohomologies of states and entanglement witnesses. Sheafification erases the global-from-local signature while leaving within-patch multipartite structure, captured by local entanglement groups introduced here. For smooth parameter families, the obstruction admits a differential-geometric representative obtained by pairing an appropriate witness field with the curvature of a natural unitary connection on the associated bundle of amplitudes. We also introduce a Quantum Entanglement Index (QEI) as an index-theoretic invariant of entangled states and explain its behavior. Finally, we outline a theoretical physics approach to probe these ideas in quantum many-body systems and suggest a possible entanglement-induced correction as an experimental validation. Detailed numerical implementations for concrete quantum many-body models are presented in the companion paper \cite{2026arXiv260113467I}.