Breakdown of a concavity property of mutual information for non-Gaussian channels

Let $S$ and $\tilde S$ be two independent and identically distributed random variables, which we interpret as the signal, and let $P_1$ and $P_2$ be two communication channels. We can choose between two measurement scenarios: either we observe $S$ through $P_1$ and $P_2$, and also $\tilde S$ through $P_1$ and $P_2$; or we observe $S$ twice through $P_1$, and $\tilde{S}$ twice through $P_2$. In which of these two scenarios do we obtain the most information on the signal $(S, \tilde S)$? While the first scenario always yields more information when $P_1$ and $P_2$ are additive Gaussian channels, we give examples showing that this property does not extend to arbitrary channels. As a consequence of this result, we show that the continuous-time mutual information arising in the setting of community detection on sparse stochastic block models is not concave, even in the limit of large system size. This stands in contrast to the case of models with diverging average degree, and brings additional challenges to the analysis of the asymptotic behavior of this quantity.