Entanglement-Rank Duality in Quadratic Phase Quantum States

• URL: http://arxiv.org/abs/2605.05167v1
• Date: Wed, 06 May 2026 17:34:42 GMT
• Title: Entanglement-Rank Duality in Quadratic Phase Quantum States
• Abstract summary: We identify a finite-field rank structure underlying multipartite entanglement in a class of quadratic-phase quantum states.<n>We show that the entanglement structure factorises via the Chinese Remainder Theorem into independent prime-field contributions.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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