Thirty-six quantum officers are entangled

TL;DR

This paper investigates the existence of mutually orthogonal quantum Latin squares of order six, proving they cannot exist without entanglement, thus highlighting the essential role of entanglement in quantum combinatorial designs.

Contribution

It demonstrates that mutually orthogonal quantum Latin squares of order six do not exist without entanglement, establishing a fundamental difference from classical Latin squares.

Findings

Mutually orthogonal quantum Latin squares of order six do not exist without entanglement.

Entanglement is necessary for the existence of quantum Latin squares of order six.

Classical Latin squares have solutions for all orders except 2 and 6, but quantum solutions differ due to entanglement.

Abstract

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose, Shrikhande and Parker, 1960]. 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 six [Rather et al., 2022]. We prove that mutually orthogonal quantum Latin squares of order six do not exist if entanglement is not allowed.