Quantum Latin Squares of Order Six with Cardinalities Nineteen, Twenty-One, and Twenty-Three

Abstract

We give three explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, and $23$, where vectors differing only by a global phase are counted as identical. The first two examples arise from normalized Schur products of columns of complex Hadamard matrices. For cardinality $19$, a Butson-type matrix over eighth roots of unity has the unique nontrivial coincidence $v_{01}=v_{25}=v_{34}$. For cardinality $21$, an explicit member of Karlsson's three-parameter family has $21$ pairwise inequivalent unordered Schur products. To exceed the symmetric Schur-product bound, we give a third, direct-sum construction based on the decomposition $\C^6=\C^4\oplusC^2$. It uses nineteen distinct rays in the four-dimensional summand and four rays in the two-dimensional summand, arranged so that every row and column is an orthonormal basis, yielding cardinality $23$. Together with our earlier constructions of cardinalities $13$, $15$, and $17$ and previously known order-six examples, these results determine every cardinality in the interval $6\leq c\leq24$, with the sole exception of the impossible value $c=7$.