  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.
