On the (Im)possibility of Non-interactive Correlation Distillation

Abstract

We study the problem of non-interactive correlation distillation (NICD). Suppose Alice and Bob each has a string, denoted by A = a ₀ a ₁...a _n − 1 and B = b ₀ b ₁...b _n − 1, respectively. Furthermore, for every k=0,1,...,n-1, (a _k ,b _k ) is independently drawn from a distribution $\mathcal{N}$, known as the “noise mode”. Alice and Bob wish to “distill” the correlation non-interactively, i.e., they wish to each apply a function to their strings, and output one bit, denoted by X and Y, such that Prob[X = Y] can be made as close to 1 as possible. The problem is, for what noise model can they succeed? This problem is related to various topics in computer science, including information reconciliation and random beacons. In fact, if NICD is indeed possible for some general class of noise models, then some of these topics would, in some sense, become straightforward corollaries.

We prove two negative results on NICD for various noise models. We prove that for these models, it is impossible to distill the correlation to be arbitrarily close to 1. We also give an example where Alice and Bob can increase their correlation with one bit of communication. This example, which may be of its own interest, demonstrates that even the smallest amount of communication is provably more powerful than no communication.