We prove a tight bound for one-shot channel simulation via a stronger version of the strong functional representation lemma: for any random variables $X,Y$, there exists a random variable $Z$ with $I(Z;X)=H(Y|X,Z)=0$ and $H(Y|Z)\le I+\log I+(1-γ)\log e-\log\log e+O(I^{-1}\log I)$, where $I=I(X;Y)$ and $γ$ is the Euler-Mascheroni constant. A more convenient bound is $H(Y|Z)<I+\log(I+2)+1$. The first bound is notable in the sense that the constant term $(1-γ)\log e-\log\log e$ is optimal, and cannot be improved for any bound stated only in terms of $I$. Hence, this is the tightest form of the strong functional representation lemma up to the third order. We also prove that the Poisson functional representation is almost optimal up to a constant gap, and provide a simple finite-blocklength achievability bound that implies the $(1/2)\log n$ redundancy of the simulation of memoryless non-singular channel. These are proved via a generalization of the functional information, in particular, the exponential functional information, which closely approximates the cost of one-shot channel simulation.

翻译：暂无翻译