Quantum Latin Squares With all Possible Cardinalities

Abstract

A quantum Latin square of order $n$ (denoted as QLS$\left(n\right)$) is an $n\times n$ array whose entries are unit column vectors from the $n$-dimensional Hilbert space ${\mathscr{H}}_n$, such that each row and column forms an orthonormal basis. Two unit vectors $\mid u〉,\mid v〉\in {\mathscr{H}}_n$ are regarded as identical if there exists a real number $\theta$ such that $\mid u〉=e^{i\theta }\mid v〉$; otherwise, they are considered distinct. The cardinality $c$ of a QLS$\left(n\right)$ is the number of distinct vectors in the array. In this paper, we use sub-QLS$\left(4\right)$s to prove that for any integer $m\ge 2$, and any integer $c\in \left[4m,16m^2\right]⧹\left\{4m+1\right\}$, there is a QLS$\left(4m\right)$ with cardinality $c$.