  Given two density matrices $\rho$ and $\sigma$, there are a number of
different expressions that reduce to the $\alpha$-R\'enyi relative entropy of
$\rho$ with respect to $\sigma$ in the classical case; i.e., when $\rho$ and
$\sigma$ commute. Only those expressions for which the Data Processing
Inequality (DPI) is valid are of potential interest as quantum divergences in
quantum information theory. Audenaert and Datta have made a conjecture on the
validity of the DPI for an interesting family of quantum generalizations of the
$\alpha$ - R\'enyi relative entropies, the $\alpha-z$ - R\'enyi relative
entropies. They and others have contributed to the partial solution of this
conjecture. We review the problem, its context, and the methods that have been
used to obtain the results that are known at present, presenting a unified
treatment of developments that have unfolded in a number of different papers.
