  Mutual information is a measure of classical and quantum correlations of
great interest in quantum information. It is also relevant in quantum many-body
physics, by virtue of satisfying an area law for thermal states and bounding
all correlation functions. However, calculating it exactly or approximately is
often challenging in practice. Here, we consider alternative definitions based
on R\'enyi divergences. Their main advantage over their von Neumann counterpart
is that they can be expressed as a variational problem whose cost function can
be efficiently evaluated for families of states like matrix product operators
while preserving all desirable properties of a measure of correlations. In
particular, we show that they obey a thermal area law in great generality, and
that they upper bound all correlation functions. We also investigate their
behavior on certain tensor network states and on classical thermal
distributions.
