Quantum Latin squares with all possible cardinalities

TL;DR

This paper demonstrates the existence of quantum Latin squares of certain sizes with all possible numbers of distinct vectors, expanding the known range of their cardinalities.

Contribution

It establishes the existence of quantum Latin squares of order multiples of four with all cardinalities in a specified range, filling gaps in the known spectrum.

Findings

Existence of QLS(4m) with cardinality c for specified ranges

Construction method using sub-QLS(4)

Completes the spectrum of possible cardinalities for certain QLSs

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 $|u\rangle, |v\rangle\in \mathcal{H}_n$ are regarded as identical if there exists a real number $\theta$ such that $|u\rangle=e^{i\theta}|v\rangle$; 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 $m\geq 2$ and any integer $c\in [4m,16m^2]\setminus \{4m+1\}$, there is a QLS$(4m)$ with cardinality $c$.