Karol Zyczkowski (Jagiellonian University Cracow, Poland Center for Theoretical Physics, PAS, Warsaw)
Joint work with S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioc and Arul Lakshminarayan
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 example of 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 special state deserves the appellation golden AME state as the golden ratio appears prominently in its elements. This state enables us to construct a pure nonadditive 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 triplet of such states.