Quantum orthogonal Latin squares of order six

Determine whether there exist two orthogonal quantum Latin squares of order six, defined as a 6×6 array of 36 orthonormal bipartite states in C^6 ⊗ C^6 whose row and column sums are maximally entangled states; equivalently, ascertain the existence or nonexistence of an absolutely maximally entangled (AME) state of four subsystems with local dimension six, a 2-unitary matrix in U(36), or a perfect four-index tensor with indices of range 1…6.

Background

Classical orthogonal Latin squares (OLS) of order six do not exist, as proven by Tarry, implying the classical Euler 36 officers problem has no solution. The quantum analogue uses orthogonal quantum Latin squares (OQLS), where entries are bipartite states and row/column superpositions are maximally entangled.

The existence of two OQLS of order six is equivalent to several structures: an AME(4,6) state, a 2-unitary U ∈ U(36), or a perfect tensor with four indices of size six. Resolving this would launch progress in quantum combinatorics and multipartite entanglement theory.

References

An analogous quantum problem, which involves 36 entangled officers, remains open.

— Five open problems in quantum information  (2002.03233 - Horodecki et al., 2020) in Section: Discrete structures in the Hilbert space; Subsection: Quantum Orthogonal Latin Squares

This leaves two natural questions: What is the smallest order $n \ge 2$ for which an $\mathrm{MCOQLS}(n)$ exists? Does there exist an integer $n_0$ such that an $\mathrm{MCOQLS}(n)$ exists for every $n\geq n_0$?

— Orthogonal Quantum Latin Squares with Maximal Cardinality from Full-Rank Factorizations  (2609.29070 - Lv et al., 24 Sep 2026) in Section 5, Conclusion