Secret, public and quantum correlation cost of triples of random variables

The inverse of Maurer's secret key distillation problem from (many independent realisations of) a triple of random variables X, Y, Z by two players (Alice and Bob) against an eavesdropper (Eve) is considered: the formation of the joint distribution (up to local degrading of Z) from secret key and public communication. We determine the asymptotically minimal amount of secret key for this task, and indeed the full trade-off of secret (between Alice and Bob) vs. public (shared between Alice, Bob and Eve) correlation for this problem. Our result generalises a theorem of Wyner on the "common information of a pair of random variables", which is recovered as the special case of Z being independent of XY. We investigate the secret key required as a function of the probability distribution and compare to an analogous notion based on prior shared entanglement