Quantum Latin squares of order $6m$ with all possible cardinalities

Abstract

A quantum Latin square of order $n$ (denoted as QLS$(n)$) is an $n\times n$ array whose entries are unit column vectors from the $n$-dimensional Hilbert space ${\mathcal{H}}_n$, such that each row and column forms an orthonormal basis. Two unit vectors $|u⟩,|v⟩\in {\mathcal{H}}_n$ are regarded as identical if there exists a real number $\theta$ such that $|u⟩=e^{i\theta }|v⟩$; otherwise, they are considered distinct. The cardinality $c$ of a QLS$(n)$ is the number of distinct vectors in the array. In this note,we use sub-QLS$(6)$ to prove that for any integer $m\ge 2$ and any $c\in [6m,36m^2]\setminus \{6m+1\}$, there is a QLS$(6m)$ with cardinality $c$.