Communicating Probability Distributions

A rate distortion problem is solved that is motivated by a quantum data compression problem. The goal is to send information about a source string x so that a receiver can construct a second string y for which the joint empirical probability distribution of x and y is close to some desired distribution. The problem differs from the usual rate distortion problems in that one must consider both remote sources and distortion functions that are not averages of per-letter distortion functions