Thirty-Six Quantum Officers Are Entangled

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose et al., Further results on the construction of mutually orthogonal latin squares and the falsity of Euler's conjecture, Can. J. Math. 12, 189 (1960)CJMAAB0008-414X10.4153/CJM-1960-016-5]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution": there exist so-called entangled quantum Latin squares of order 6 [Rather et al., Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Phys. Rev. Lett. 128, 080507 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.080507]. We prove that mutually orthogonal quantum Latin squares of order 6 do not exist if entanglement is not allowed.