Communicating Probability Distributions as Compressed Quantum Information

A rate distortion problem is considered where the goal is to send information about a source sequence so that the receiver can construct a second sequence for which the joint empirical probability distribution of the two sequences is close to some desired distribution. This problem arises in two scenarios involving the efficient transmission of quantum information: compressing quantum states and compressing the outcomes of quantum measurements. The problem differs from the usual rate distortion problems in that one must consider "remote" sources and sequence distortions that are not averages of single letter distortion functions.