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.