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Ordering and classifying multipartite quantum states by their entanglementcontent remains an open problem. One class of highly entangled states, usefulin quantum information protocols, the absolutely maximally entangled (AME)ones, are specially hard to compare as all their subsystems are maximallyrandom. While, it is well-known that there is no AME state of four qubits, manyanalytical examples and numerically generated ensembles of four qutrit AMEstates are known. However, we prove the surprising result that there is trulyonly {\em one} AME state of four qutrits up to local unitary equivalence. Incontrast, for larger local dimensions, the number of local unitary classes ofAME states is shown to be infinite. Of special interest is the case of localdimension 6 where it was established recently that a four-party AME state doesexist, providing a quantum solution to the classically impossible Euler problemof 36 officers. Based on this, an infinity of quantum solutions are constructedand we prove that these are not equivalent. The methods developed can beusefully generalized to multipartite states of any number of particles.

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