A Hodge Theory for Entanglement Cohomology

• URL: http://arxiv.org/abs/2410.12529v3
• Date: Thu, 26 Jun 2025 15:18:46 GMT
• Title: A Hodge Theory for Entanglement Cohomology
• Authors: Christian Ferko, Eashan Iyer, Kasra Mossayebi, Gregor Sanfey,
• Abstract summary: We explore and extend the application of homological algebra to describe quantum entanglement.<n>We construct analogues of the Hodge star operator, inner product, codifferential, and Laplacian for entanglement $k$-forms.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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