Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler

TL;DR

This paper classifies absolutely maximally entangled states, proving uniqueness for four qutrits and infinitely many for larger dimensions, and constructs infinite quantum solutions to the Euler problem of 36 officers.

Contribution

It establishes the uniqueness of four qutrit AME states and constructs an infinite family of inequivalent AME states for larger dimensions, including solutions to the Euler problem.

Findings

Only one four-qutrit AME state up to local unitaries.

Infinite local unitary classes of AME states for dimensions greater than four.

Constructed infinite quantum solutions to the Euler problem of 36 officers.

Abstract

Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are specially hard to compare as all their subsystems are maximally random. While, it is well-known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four qutrit AME states are known. However, we prove the surprising result that there is truly only {\em one} AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6 where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.