Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem

TL;DR

This paper demonstrates that entanglement enables solutions to classical combinatorial problems, specifically constructing quantum Latin squares and maximally entangled states that surpass classical limitations.

Contribution

It introduces the first orthogonal quantum Latin squares of size six and constructs a novel maximally entangled state, the golden AME, with applications in quantum error correction.

Findings

Constructed orthogonal quantum Latin squares of order six.

Created the golden AME(4,6) state with maximal entangling power.

Developed a quantum error detection code saturating the Singleton bound.

Abstract

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 example of the long-elusive 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 result allows 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.