Distributed Source Simulation With No Communication

TL;DR

This paper investigates the conditions under which distributed source simulation without communication can approximate a target joint distribution, providing a single-letter criterion and exploring special cases with binary sources.

Contribution

It introduces a single-letter condition for asymptotic distributed source simulation without communication and examines the special case of binary sources, proposing a conjecture about scalar Markov chain simulation.

Findings

A single-letter condition for asymptotic simulation with vanishing KL divergence.

The condition is nontrivial when Gács-Körner common information is nonzero.

Close to the boundary case, simulations approximate scalar Markov chains in total variation.

Abstract

We consider the problem of distributed source simulation with no communication, in which Alice and Bob observe sequences $U^n$ and $V^n$ respectively, drawn from a joint distribution $p_{UV}^{\otimes n}$, and wish to locally generate sequences $X^n$ and $Y^n$ respectively with a joint distribution that is close (in KL divergence) to $p_{XY}^{\otimes n}$. We provide a single-letter condition under which such a simulation is asymptotically possible with a vanishing KL divergence. Our condition is nontrivial only in the case where the G\`acs-K\"orner (GK) common information between $U$ and $V$ is nonzero, and we conjecture that only scalar Markov chains $X-U-V-Y$ can be simulated otherwise. Motivated by this conjecture, we further examine the case where both $p_{UV}$ and $p_{XY}$ are doubly symmetric binary sources with parameters $p,q\leq 1/2$ respectively. While it is trivial that in this case $p\leq q$ is both necessary and sufficient, we show that when $p$ is close to $q$ then any successful simulation is close to being scalar in the total variation sense.