  We explore a large class of correlation measures called the $\alpha-z$
R\'enyi mutual informations (RMIs). Unlike the commonly used notion of RMI
involving linear combinations of R\'enyi entropies, the $\alpha-z$ RMIs are
positive semi-definite and monotonically decreasing under quantum operations,
making them sensible measures of total (quantum and classical) correlations.
This follows from their descendance from R\'enyi relative entropies. In
addition to upper bounding connected correlation functions between subsystems,
we prove the much stronger statement that for certain values of $\alpha$ and
$z$, the $\alpha-z$ RMIs also lower bound connected correlation functions. We
develop an easily implementable replica trick which enables us to compute the
$\alpha-z$ RMIs in a variety of many-body systems including conformal field
theories, free fermions, random tensor networks, and holography.
