  The existence of a global causal order between events places constraints on
the correlations that parties may share. Such "causal correlations" have been
the focus of recent attention, driven by the realization that some extensions
of quantum mechanics may violate so-called causal inequalities. In this paper
we study causal correlations from an entropic perspective, and we show how to
use this framework to derive entropic causal inequalities. We consider two
different ways to derive such inequalities. Firstly, we consider a method based
on the causal Bayesian networks describing the causal relations between the
parties. In contrast to the Bell-nonlocality scenario, where this method has
previously been shown to be ineffective, we show that it leads to several
interesting entropic causal inequalities. Secondly, we consider an alternative
method based on counterfactual variables that has previously been used to
derive entropic Bell inequalities. We compare the inequalities obtained via
these two methods and discuss their violation by noncausal correlations. As an
application of our approach, we derive bounds on the quantity of information -
which is more naturally expressed in the entropic framework - that parties can
communicate when operating in a definite causal order.

Quantum Physics