Entanglement-Rank Duality in Quadratic Phase Quantum States

TL;DR

This paper reveals a finite-field rank structure in quadratic-phase quantum states, establishing a duality between entanglement and matrix rank, and enabling systematic construction of highly entangled states.

Contribution

It introduces an exact Rank-Purity Duality linking subsystem purity to phase matrix rank, facilitating the systematic creation of maximally entangled states.

Findings

Subsystem R'eni-2 purity is determined by phase matrix rank.

Existence of AME states is equivalent to full-rank bipartition submatrices.

Entanglement structure factorises into prime-field contributions for square-free dimensions.

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 R\'enyi-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 R\'enyi-2 entropies. These results provide an algebraic characterisation of entanglement in the quadratic phase formalism and enable the systematic construction of highly entangled states.