New Cardinalities for Quantum Latin Squares of Order Six

Abstract

We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, $27$, $32$, and $35$, where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$, these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$.