I have been working on privacy and come across a neat problem.Suppose two random variables $X$ and $Y$, over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$, are given with joint distribution $P_{XY}$.We want to make sure that there exists a non-degenerate function acting on $\mathcal{Y}$, say, $f(Y)$, such that $f(Y)$ is independent of $X$.I am looking for a sufficient and necessary condition of this.It is clear that a necessary condition for this is that $P_{XY}$ be rank deficient. I am, in particular, hoping to see if there is any connection to the set of singular values of  the conditional expectation operator (like the connection between maximal correlation and the second largest singular value of conditional expectation operator).
One kind of dependence relation between a pair of random variables
cr.crypto security;pr.probability;it.information theory;st.statistics