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Entanglement Theory Quantum Correlations Open Quantum Systems Decoherence Quantum Simulation

Hodge Theory for Entanglement Cohomology

arXiv
Authors: Christian Ferko, Eashan Iyer, Kasra Mossayebi, Gregor Sanfey

Year

2024

Paper ID

38043

Status

Preprint

Abstract Read

~2 min

Abstract Words

104

Citations

N/A

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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  • 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...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2410.12529 [2] arXiv https://arxiv.org/abs/2410.12529 [3] Publisher https://arxiv.org/abs/2410.12529

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