On Non-Existence of Stabilizer Absolutely Maximally Entangled States in Even Local Dimensions

Abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of $N=4n$ qudits with $n\in\{1,2,3,...\}$, each of even local dimension, cannot be realized as graph states. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. In particular, this study provides an independent solution of the recently discussed case of the AME state of four quhexes and clarifies its characterization within the stabilizer formalism, complementing the results found recently in [H. Cha, arXiv:2603.13442]. At the same time, we provide a general construction for mixed $k$-uniform states whose purity is determined by the optimal stabilizer representations. For the specific case of $(N=4,d=6)$, this yields a mixed AME state of optimal purity $1/2$, not subject to canonical graph-state constraints.