Existence of an $\operatorname{AME}(11,5)$ graph state

The problem asks whether an absolutely maximally entangled state with $11$ parties of local dimension $5$ can be realized as a graph state. The available literature gives a criterion for graph-state realizations, but no construction or obstruction specific to this case.

Known results

• Graph-state AME status is characterized by linear independence conditions on submatrices of the adjacency matrix over $Z_p$ (2013).
• General classification of $\operatorname{AME}(n,d)$ states remains open in higher dimensions.

August 2026 existence report

A paper’s existence table lists $\operatorname{AME}(11,5)$ as existing, but attributes the entry to an earlier reference rather than presenting a new construction. Its explicit constructions include $\operatorname{AME}(12,5)$, not $\operatorname{AME}(11,5)$, and it supplies no graph-state-specific proof.

Current status (as of August 2026): existence of a general $\operatorname{AME}(11,5)$ state is reported, while existence of an $\operatorname{AME}(11,5)$ graph state remains open.