Source Coding for Synthesizing Correlated Randomness

We consider a scenario wherein two parties Alice and Bob are provided $X_{1}^{n}$ and $X_{2}^{n}$ samples that are IID from a PMF $p_{X_1 X_2}$. Alice and Bob can communicate to Charles over (noiseless) communication links of rate $R_1$ and $R_2$ respectively. Their goal is to enable Charles generate samples $Y^{n}$ such that the triple $(X_{1}^{n},X_{2}^{n},Y^{n})$ has a PMF that is close, in total variation, to $\prod p_{X_1 X_2 Y}$. In addition, the three parties may posses shared common randomness at rate $C$. We address the problem of characterizing the set of rate triples $(R_1,R_2,C)$ for which the above goal can be accomplished. We provide a set of sufficient conditions, i.e., an achievable rate region for this three party setup. Our work also provides a complete characterization of a point-to-point setup wherein Bob is absent and Charles is provided with side-information.