Fine asymptotics for approximate simulation of symmetric channels

We study the approximate channel simulation problem in the normal approximation regime. Our starting point is the fact established by Cao et al.~\cite{Cao_channel_simulation}, that the minimal codebook size $M_{\epsilon }^{\ast }(W)$ for a fixed error tolerance $\epsilon >0$ is closely related to the $\epsilon$-smooth channel max-information, $I_{\max }^{}(W)$. Using this, we refine earlier large-blocklength approximations to $M_{\epsilon }^{\ast }(W^{\otimes n})$ for discrete memoryless channels. Specifically, we provide tight third-order expansions for Gallager-symmetric channels and demonstrate that the third-order term is channel-dependent. As in corresponding channel coding expansions, we identify a substantial rate gap between singular and nonsingular channels. Notably, the direction of this gap is reversed relative to channel coding: The third-order term is $-\frac{1}{2}\log n$ for singular channels but $O(\log \log n)$ for nonsingular channels.