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

TL;DR

This paper proves that certain highly entangled quantum states called AME states cannot be represented as graph states when the local dimension is even, and provides new constructions for mixed states with optimal purity.

Contribution

It establishes non-existence results for AME states as graph states in even dimensions and introduces a construction for mixed states with optimal stabilizer-based purity.

Findings

AME states with even local dimensions cannot be realized as graph states.

Provides a construction for mixed k-uniform states with optimal stabilizer-based purity.

Yields a mixed AME state of optimal purity 1/2 for (N=4, d=6).

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.