Say we have two random variables $X$, $Y$. They are discrete variables (or discretization of continuous variables), both with $k$ categories. Define the left stochastic matrix as $P(X|Y)_{ij}:=p(x_i|y_j)$ with shape $k \times k$.
My question is what's the equivalent condition for $P(X|Y)$ to be invertible? Does the invertibility need any constraints on the probability distribution of $p(x)$, $p(y)$, or $p(x,y)$?
I found some similar questions here and here. However, neither of them answers my question.
References, if any, are appreciated.