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 $d\geq 3$.

arXiv:2608.01011 [quant-ph]

(or arXiv:2608.01011v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.01011 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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