Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

Abstract

We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.

The paper addresses the existence of seven-party absolutely maximally entangled states, $\mathrm{AME}(7,d)$, whose reductions to any three parties are maximally mixed. It proves that such states exist exactly when $d\geq3$, under the usual convention excluding $d=1$. This completes the classification using the previously established nonexistence at $d=2$, and supplies explicit constructions for previously unresolved dimensions, including $d=6$ and $d=10$. For every odd dimension $q\geq3$, a quadratic-phase graph state is constructed from the complement of the seven-cycle. Maximal mixing follows from invertible cross-cut minors: cyclic symmetry reduces the 35 three-party cuts to five representatives, with selected determinants $-1,1,2,-1,1$, all invertible modulo odd $q$. For $d=2q$ with odd $q\geq3$, the construction coherently couples the two logical states of the seven-qubit Steane code to locally phase-shifted versions of the odd-dimensional graph state. Off-diagonal reduced blocks vanish in the binary sector on 28 cuts. On the remaining seven Fano-plane cuts, explicit left-null-vector certificates establish vanishing in the odd-dimensional sector. The diagonal blocks combine to give $I_{d^3}/d^3$ on every cut. Known power-of-two constructions and the tensor-product property then cover all remaining dimensions. The classification is specific to seven parties. Together with existing results, it completes the existence classification through seven parties; the paper identifies $\mathrm{AME}(8,4)$ as the first remaining open case and leaves classification for larger party numbers unresolved.