Sufficient conditions for the equality of exact and Wyner common information

Abstract

The minimum common randomness required for the approximate and separate generation of a pair of correlated discrete memoryless sources is quantified by Wyner's notion of common information. Recently, Kumar, Li, and El Gamal introduced the notion of exact common information as the minimum common randomness required for the exact and separate generation of a pair of correlated discrete memoryless sources. This new notion of common information, which does not have a general single-letter characterization, was shown to match Wyner's notion for the symmetric binary erasure source. In this work, we present two conditions on the joint statistics of the pair of sources under either of which the exact and Wyner's notions of common information coincide. Though the conditions are implicit, we prove the equality of Wyner and exact common information for the generalized binary Z-source, generalized erasure source and the noisy typewriter source by establishing that these sources meet either of these conditions.