Let $X$,$Y$ be two stochastic variables with probability distribution $\rho(X,Y)$. The
mutual information, $I(X;Y)$, represents the information shared by the two variables. This intuitive interpretation of $I(X;Y)$ suggests the following question. Given $\rho(X,Y)$, is there a process
$X\rightarrow Z\rightarrow Y$ so that the entropy of $Z$ is equal to the mutual information $I(X;Y)$? 
Indeed, the minimal amount of information contained in $Z$ should be the information strictly necessary for reproducing the correlations between $X$ and $Y$. It is easy to show that $E(Z)\ge I(X;Y)$, but it is not clear to me if the strict equality can be reached for any $\rho(X,Y)$.



mutual information and minimal communication required for generating correlation
I have found this paper that is closely related with my question:


"The communication complexity of Correlation" by P. Harsha, R. Jain, D. McAllester, J. Radhakrishnan, IEEE Transactions on Information Theory 56, 438 (2010).


In this paper, it is shown that the communication cost required for generating
correlation can be very close to the mutual information. Indeed, it is proved
that the communication cost is smaller or equal to the mutual information, I, plus
2 log(I+1). The authors consider a scenario where random resources are shared
by the parties. 


