Entanglement-Rank Duality in Quadratic Phase Quantum States
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arXiv
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| Format: | Preprint |
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2026
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| author | Dahbi, Zakaria Zair, Amelle |
| author_facet | Dahbi, Zakaria Zair, Amelle |
| contents | Absolutely Maximally Entangled (AME) states are important resources in quantum information processing; however, a general systematic approach for constructing these states remains a formidable challenge. We identify a finite-field rank structure underlying multipartite entanglement in a class of quadratic-phase quantum states defined by symmetric matrices over $\mathbb{F}_p$. We prove an exact Rank-Purity Duality: the Rényi-2 purity of any subsystem is determined solely by the rank of the phase matrix. Within this ansatz, the existence of an AME state is equivalent to the existence of a generating phase matrix $P$ whose bipartition submatrices are of full rank, reducing the condition for maximal entanglement to a rank constraint on $P$. This establishes a direct correspondence between entanglement and cut-rank geometry in finite-field matrices. Furthermore, for square-free local dimensions, we show that the entanglement structure factorises via the Chinese Remainder Theorem into independent prime-field contributions, yielding an exact additive decomposition of Rényi-2 entropies. These results provide an algebraic characterisation of entanglement in the quadratic phase formalism and enable the systematic construction of highly entangled states. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Entanglement-Rank Duality in Quadratic Phase Quantum States Dahbi, Zakaria Zair, Amelle Quantum Physics Absolutely Maximally Entangled (AME) states are important resources in quantum information processing; however, a general systematic approach for constructing these states remains a formidable challenge. We identify a finite-field rank structure underlying multipartite entanglement in a class of quadratic-phase quantum states defined by symmetric matrices over $\mathbb{F}_p$. We prove an exact Rank-Purity Duality: the Rényi-2 purity of any subsystem is determined solely by the rank of the phase matrix. Within this ansatz, the existence of an AME state is equivalent to the existence of a generating phase matrix $P$ whose bipartition submatrices are of full rank, reducing the condition for maximal entanglement to a rank constraint on $P$. This establishes a direct correspondence between entanglement and cut-rank geometry in finite-field matrices. Furthermore, for square-free local dimensions, we show that the entanglement structure factorises via the Chinese Remainder Theorem into independent prime-field contributions, yielding an exact additive decomposition of Rényi-2 entropies. These results provide an algebraic characterisation of entanglement in the quadratic phase formalism and enable the systematic construction of highly entangled states. |
| title | Entanglement-Rank Duality in Quadratic Phase Quantum States |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.05167 |