On the Existence of Absolutely Maximally Entangled States of Minimal Support II

TL;DR

This paper investigates the existence of absolutely maximally entangled states with minimal support, establishing conditions based on coding theory and Latin hypercubes, and demonstrating their failure in certain non-prime power dimensions.

Contribution

It introduces a new approach using Latin hypercubes to analyze AME states and shows the non-existence in specific cases where previous conditions suggested possible existence.

Findings

AME states of minimal support are linked to coding theory and Latin hypercubes.

The sufficient condition for existence fails when the local dimension is not a prime power, e.g., d=6.

Results extend understanding of the existence conditions for AME states in various dimensions.

Abstract

Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimension $d$ is a power of a prime number. In this paper, an equivalence with latin hypercubes is used to prove that the above sufficient condition fails in the first case in which the local dimension is not a prime power, $d=6$. Results for other values of $d$ are also given.