  The main result in this article shows that the quantum $f$-divergence of two
states is equal to the classical $f$-divergence of the corresponding
Nussbaum-Szko{\l}a distributions. This provides a general framework for
studying certain properties of quantum entropic quantities using the
corresponding classical entities. The usefulness of the main result is
illustrated by obtaining several quantum $f$-divergence inequalities from their
classical counterparts. All results presented here are valid in both finite and
infinite dimensions and hence can be applied to continuous variable systems as
well. A comprehensive review of the instances in the literature where
Nussbaum-Szko{\l}a distributions are used, is also provided in this article.
