A Quantum Latin Square of Order Six with Cardinality 29

Abstract

We construct an explicit real quantum Latin square of order six with cardinality $29$, the last unresolved value in the order-six spectrum. We first construct a punctured $6 \times 6$ array in $\mathbb{R}^5$ whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order $6$.