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

Abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of $N=4n$ qudits with $n\in \{1,2,3,\dots \}$, each of even local dimension $d$, cannot be realized as canonical graph states over the ring ${\mathbb{Z}}_d$. 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. Furthermore, for $d\equiv 2(\mathrm{mod}4)$ this obstruction, combined with the prime-power decomposition of stabilizer states, excludes all pure stabilizer AME states (including four quhexes), while clarifying the distinction between canonical ${\mathbb{Z}}_d$ graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed $k$-uniform states. For $(N,d)=(4,6)$, the construction yields a rank-two mixed $2$-uniform stabilizer state of purity $1/2$, which is optimal among normalized stabilizer projectors.

arXiv comment5 pages