On Non-Interactive Simulation of Joint Distributions

Abstract

We consider the following non-interactive simulation problem: Alice and Bob observe sequences $X^{n}$ and $Y^{n}$ , respectively, where $\{(X_{i}, Y_{i})\}_{i=1}^{n}$ are drawn independent identically distributed from $P(x,y)$ , and they output $U$ and $V$ , respectively, which is required to have a joint law that is close in total variation to a specified $Q(u,v)$ . It is known that the maximal correlation of $U$ and $V$ must necessarily be no bigger than that of $X$ and $Y$ if this is to be possible. Our main contribution is to bring hypercontractivity to bear as a tool on this problem. In particular, we show that if $P(x,y)$ is the doubly symmetric binary source, then hypercontractivity provides stronger impossibility results than maximal correlation. Finally, we extend these tools to provide impossibility results for the $k$ -agent version of this problem.