Hodge Theory for Entanglement Cohomology

TL;DR

This paper applies Hodge theory to entanglement cohomology, establishing a geometric framework that reveals symmetries and dualities in quantum entanglement structures.

Contribution

It develops Hodge-theoretic tools for entanglement cohomology, including operators and isomorphisms, and proves duality properties in finite-dimensional quantum systems.

Findings

Hodge star, inner product, codifferential, Laplacian analogues constructed for entanglement forms

Hodge isomorphism and decomposition theorems proven for entanglement cohomology

Symmetry property (Poincare duality) of cohomology dimensions established

Abstract

We explore and extend the application of homological algebra to describe quantum entanglement, initiated in arXiv:1901.02011, focusing on the Hodge-theoretic structure of entanglement cohomology in finite-dimensional quantum systems. We construct analogues of the Hodge star operator, inner product, codifferential, and Laplacian for entanglement $k$-forms. We also prove that such $k$-forms obey versions of the Hodge isomorphism theorem and Hodge decomposition, and that they exhibit Hodge duality. As a corollary, we conclude that the dimensions of the $k$-th and $(n-k)$-th cohomologies coincide for entanglement in $n$-partite pure states, which explains a symmetry property ("Poincare duality") of the associated Poincare polynomials.