Distributed Source Simulation With No Communication

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