TITLE: Variant of mutual information
QUESTION [9 upvotes]: Given a discrete random variable $(X,Y)$, one can consider the smallest entropy of a random variable $Z$ such that $X$ and $Y$ are independent conditioned to $Z$. 
This quantity is akin to the mutual information, in fact one can see that it is always larger than half the mutual information.
Does this number have a name, and has it been studied?

REPLY [5 votes]: This is related to Wyner's common information which is always larger than the mutual information.
https://ieeexplore.ieee.org/document/1055346/