Abstract
A quantum Latin square of order n (denoted as QLS(n)) is an n× n array whose entries are unit column vectors from the n-dimensional Hilbert space H_n, such that each row and column forms an orthonormal basis. Two unit vectors |u, |v∈ H_n are regarded as identical if there exists a real number θ such that |u=e^iθ|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≥ 2 and any c∈ [6m,36m²] {6m+1}, there is a QLS(6m) with cardinality c.