Suppose we are given two random variables $X$ and $Y$ with fixed marginal and joint distribution. What is the maximum randomness that we can extract from $Y$ that is independent from $X$, that is, if $Z=f(Y)$ is a transformation of $Y$, what is $\sup_f H(Z)$ where supremum is taken over all functions $f$ for which $f(Y)$ and $X$ are independent.
A really loose bound for this is $H(Y|X)$. This is loose, because for example if $X$ and $Y$ are dependent binary random variables with some joint distribution, independence of $Z$ and $X$ yields independence of $Z$ and $Y$, however, $H(Y|X)$ is not necessarily not zero.
A natural extension of this, is to relax the condition of "$Z$ and $X$ being independent" to "$Z$ and $X$ being $\epsilon$-independent", which means, mutual information $I(Z;X)\leq \epsilon$ or $|P(Z, X)-P(Z)P(V)|\leq\epsilon$ or any other notion of $\epsilon$-independence that you can think of for making this problem easier..
I would be really thankful if you can give any idea on this,
