A Criterion of Extractability of the Mutual Information for a Triple of Strings

We say that the mutual information of a triple of binary strings $a$, $b$, $c$ can be extracted if there exists a string $d$ such that $a$, $b$, and $c$ are independent given $d$, and $d$ is simple conditional to each of the strings $a$, $b$, and $c$. It is proved that the mutual information between $a$, $b$, and $c$ can be extracted if and only if the values of the conditional mutual informations $I(a:b|c)$, $I(a:c|b)$, and $I(b:c|a)$ are negligible. The proof employs a non-Shannon-type information inequality (a generalization of the recently discovered Zhang–Yeung inequality).