On Non-Interactive Simulation of Joint Distributions

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 i.i.d. 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.