Hodge Theory for Entanglement Cohomology

TL;DR

The paper develops a Hodge-theoretic framework for entanglement cohomology in finite-dimensional multipartite quantum systems by extending the commutant entanglement complex with a Hodge star, inner product, and Laplacian. It proves a Hodge isomorphism and decomposition, establishing a duality that enforces symmetry between $k$-th and $(n-k)$-th entanglement cohomology groups and hence symmetry of the associated Poincaré polynomials. The approach relies on Schmidt-based constructions and a wedge-like shuffle product to define a consistent differential graded algebra of entanglement forms. Concrete two-qubit examples illustrate how cohomology distinguishes entangled from product states and demonstrate harmonic representatives and duality. The results provide a rigorous, symmetry-respecting framework for classifying entanglement patterns and point to extensions to mixed states, purification, and quantum field theoretic contexts.

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.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.1: Hodge isomorphism
• proof
• Theorem 3.2: Hodge decomposition
• proof