Conditional Probability Matrix and the $S^2$-rank

Using the $det^{S^2}$ map from [5], we introduce the notion of $S^2$-rank of a matrix of type $d\times \frac{s(s-1)}{2}$. As an application, we show that the conditional probability matrix associated to two random variables has the $S^2$-rank equal to $1$. Under suitable conditions we prove that the converse of this result also holds.