Quantum Latin squares 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 $\hat{I}¸$ such that $|u⟩=e^{i\hat{I}¸}|v⟩$; otherwise, they are considered distinct. The cardinality $c$ of a QLS$(n)$ is the number of distinct vectors in the array. In this paper, we use sub-QLS$(4)$s to prove that for any integer $m\ge 2$ and any integer $c\in [4m,16m^2]\setminus \{4m+1\}$, there is a QLS$(4m)$ with cardinality $c$.