  In 2000, Evans et al. [Eva+00] proved the subadditivity of the mutual
information in the broadcasting on tree model with binary vertex labels and
symmetric channels. They raised the question of whether such subadditivity
extends to loopy graphs in some appropriate way. We recently proposed such an
extension that applies to general graphs and binary vertex labels [AB18], using
synchronization models and relying on percolation bounds. This extension
requires however the edge channels to be symmetric on the product of the
adjacent spins. A more general version of such a percolation bound that applies
to asymmetric channels is also obtained in [PW18], relying on the SDPI, but the
subadditivity property does not follow with such generalizations.
  In this note, we provide a new result showing that the subadditivity property
still holds for arbitrary (asymmetric) channels acting on the product of spins,
when the graphs are restricted to be series-parallel. The proof relies on the
use of the Chi-squared mutual information rather than the classical mutual
information, and various properties of the former are discussed.
  We also present a generalization of the broadcasting on tree model (the
synchronization on tree) where the bound from [PW18] relying on the SPDI can be
significantly looser than the bound resulting from the Chi-squared
subadditivity property presented here.
