Exact Channel Synthesis
We consider the exact channel synthesis problem, which is the problem of determining how much information is required to create exact correlation remotely when there is a certain amount of randomness shared by two terminals. This problem generalizes an existing approximate version, in which the generated joint distribution is restricted to be close to a target distribution under the total variation (TV) distance measure, instead being exactly equal to the target distribution. We study the admissible region of the shared randomness rate and the communication rate for the exact channel synthesis problem by providing single-letter inner and outer bounds on it. The inner bound implies that a linear number of bits (or a finite rate) of shared randomness suffices to realize exact channel synthesis, even when communication rates are restricted to approach to the optimal communication rate I(X;Y) asymptotically. This disproves a conjecture posed by Bennett-Devetak-Harrow-Shor-Winter (2014), where they conjectured that under this scenario, an exponential number of bits of shared randomness is necessary to realize exact channel synthesis. Furthermore, our bounds coincide for doubly symmetric binary sources (or binary symmetric channels), which implies that for such sources, the admissible rate region is completely characterized. We observe that for such sources, the admissible rate region for exact channel synthesis is strictly larger than that for TV-approximate version. We also extend these results to other sources, including Gaussian sources.
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