Entanglement cohomology for GHZ and W states

TL;DR

This paper develops a cohomological framework to classify and analyze multipartite entanglement, deriving explicit dimensions for GHZ and W states, and proposing new invariants for distinguishing entanglement patterns.

Contribution

It provides exact formulas for entanglement cohomology groups of GHZ and W states and introduces novel local unitary invariants based on Laplacian spectra and intersection numbers.

Findings

Exact cohomology dimensions for GHZ and W states

Introduction of Laplacian spectrum as an entanglement invariant

Proposal of intersection numbers as new entanglement descriptors

Abstract

Entanglement cohomology assigns a graded cohomology ring to a multipartite pure state, providing homological invariants that are stable under local unitaries and characterize inequivalent patterns of entanglement. In this work we derive exact expressions for the dimensions of these cohomology groups in two canonical entanglement classes, generalized GHZ and W states on an arbitrary number of parties and local Hilbert space dimensions, thus proving conjectures of arXiv:1901.02011. Using the additional structure of the Hodge star and wedge product operations, we propose two new classes of local unitary invariants: the spectrum of the natural Laplacian acting on entanglement $k$-forms, and the intersection numbers obtained from wedge products of representatives for cohomology classes. We present numerical experiments which investigate these invariants in particular states, suggesting that they may provide useful quantities for describing multipartite entanglement.