  Bounds on information combining are entropic inequalities that determine how
the information, or entropy, of a set of random variables can change when they
are combined in certain prescribed ways. Such bounds play an important role in
information theory, particularly in coding and Shannon theory. The arguably
most elementary kind of information combining is the addition of two binary
random variables, i.e. a CNOT gate, and the resulting quantities are
fundamental when investigating belief propagation and polar coding. In this
work we will generalize the concept to R\'enyi entropies. We give optimal
bounds on the conditional R\'enyi entropy after combination, based on a certain
convexity or concavity property and discuss when this property indeed holds.
Since there is no generally agreed upon definition of the conditional R\'enyi
entropy, we consider four different versions from the literature. Finally, we
discuss the application of these bounds to the polarization of R\'enyi
entropies under polar codes.
