  Consider a pair of input distributions which after passing through a Poisson
channel become $\epsilon$-close in total variation. We show that they must
necessarily then be $\epsilon^{0.5+o(1)}$-close after passing through a
Gaussian channel as well. In the opposite direction, we show that distributions
inducing $\epsilon$-close outputs over the Gaussian channel must induce
$\epsilon^{1+o(1)}$-close outputs over the Poisson. This quantifies a
well-known intuition that ''smoothing'' induced by Poissonization and Gaussian
convolution are similar. As an application, we improve a recent upper bound of
Han-Miao-Shen'2021 for estimating mixing distribution of a Poisson mixture in
Gaussian optimal transport distance from $n^{-0.1 + o(1)}$ to $n^{-0.25 +
o(1)}$.
