Remote Channel Synthesis

We consider the problem of synthesizing a memoryless channel between an unobserved source and a remote terminal. An encoder has access to a partial or noisy version $Z^n = (Z_1, \ldots, Z_n)$ of a remote source sequence $X^n = (X_1, \ldots, X_n),$ with $(X_i,Z_i)$ independent and identically distributed with joint distribution $q_{X,Z}.$ The encoder communicates through a noiseless link to a decoder which aims to produce an output $Y^n$ coordinated with the remote source; that is, the total variation distance between the joint distribution of $X^n$ and $Y^n$ and some i.i.d. target distribution $q_{X,Y}^{\otimes n}$ is required to vanish as $n$ goes to infinity. The two terminals may have access to a source of rate-limited common randomness. We present a single-letter characterization of the optimal compression and common randomness rates. We also show that when the common randomness rate is small, then in most cases, coordinating $Z^n$ and $Y^n$ using a standard channel synthesis scheme is strictly sub-optimal. In other words, schemes for which the joint distribution of $Z^n$ and $Y^n$ approaches a product distribution asymptotically are strictly sub-optimal.