Non-existence of stabilizer absolutely maximally entangled states across infinitely many configurations

TL;DR

This paper proves that stabilizer absolutely maximally entangled states cannot exist across infinitely many configurations by reducing the problem to prime-power factors, showing that obstructions at prime levels prevent their existence in composite dimensions.

Contribution

It introduces a reduction theorem linking the existence of stabilizer AME states in composite dimensions to their prime-power factors, establishing non-existence results.

Findings

Obstructions at prime-power factors prevent stabilizer AME states in composite dimensions.

Existence of stabilizer AME states in composite dimensions depends on prime-power factors.

Theorem reduces the problem to prime-power cases, simplifying non-existence proofs.

Abstract

We prove a general reduction theorem for stabilizer absolutely maximally entangled states in composite local dimension. If a stabilizer $\mathrm{AME}(n,D)$ state exists and $D=\prod_{i=1}^m q_i$ is the prime-power factorization of $D$, then for every nonempty subset of factors there exists a stabilizer $\mathrm{AME}\bigl(n,\prod_{i\in M} q_i\bigr)$ state. Thus any obstruction at a prime-power factor immediately obstructs stabilizer AME states in the composite dimension.