Thirty-six entangled officers of Euler

The negative solution to the famous problem of $36$ officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a $2$-unitary matrix of size $36$, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This result allows us to construct a pure non-additive quhex quantum error detection code $(\!(3,6,2)\!)_6$, which saturates the Singleton bound and allows one to encode a $6$-level state into a triple of such states.