  Classical and quantum physics impose different constraints on the joint
probability distributions of observed variables in a causal structure. These
differences mean that certain correlations can be certified as non-classical,
which has both foundational and practical importance. Rather than working with
the probability distribution itself, it can instead be convenient to work with
the entropies of the observed variables. In the Bell causal structure with two
inputs and outputs per party, a technique that uses entropic inequalities is
known that can always identify non-classical correlations. Here we consider the
analogue of this technique in the generalization of this scenario to more
outcomes. We identify a family of non-classical correlations in the Bell
scenario with two inputs and three outputs per party whose non-classicality
cannot be detected through the direct analogue of the previous technique. We
also show that use of Tsallis entropy instead of Shannon entropy does not help
in this case. Furthermore, we give evidence that natural extensions of the
technique also do not help. More precisely, our evidence suggests that even if
we allow the observed correlations to be post-processed according to a natural
class of non-classicality non-generating operations, entropic inequalities for
either the Shannon or Tsallis entropies cannot detect the non-classicality, and
hence that entropic inequalities are generally not sufficient to detect
non-classicality in the Bell causal structure. In addition, for the bipartite
Bell scenario with two inputs and three outputs we find the vertex description
of the polytope of non-signalling distributions that satisfy all of the
CHSH-type inequalities, which is one of the main regions of investigation in
this work.
