Extracting the mutual information for a triple of binary 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, c. This is an analog of the well-known Gacs-Korner (1973) definition of extrability of the mutual information for a pair of binary strings. We prove that (in contrast to the case of two strings) there exists a criterion of extrability of the mutual information for a triple a, b, c in terms of complexities involving a, b, c. Roughly speaking, the mutual information between a, b, c can be extracted if and only if the conditional mutual informations I(a:b|c), I(a:c|b), I(b:c|a) are negligible. Our proof of the main result is based on a nonShannon-type information inequality, which is a generalization of the recently discovered Zhang-Yeung inequality.