Hodge Theory for Entanglement Cohomology
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866911333063065600 |
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| author | Ferko, Christian Iyer, Eashan Mossayebi, Kasra Sanfey, Gregor |
| author_facet | Ferko, Christian Iyer, Eashan Mossayebi, Kasra Sanfey, Gregor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hodge Theory for Entanglement Cohomology Ferko, Christian Iyer, Eashan Mossayebi, Kasra Sanfey, Gregor High Energy Physics - Theory Mathematical Physics Algebraic Topology Quantum Physics 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. |
| title | Hodge Theory for Entanglement Cohomology |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Topology Quantum Physics |
| url | https://arxiv.org/abs/2410.12529 |