Abstract
Given two density matrices rho and sigma, there are a number of different expressions that reduce to the alpha-Renyi 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-Renyi relative entropies, the alpha - z-Renyi 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.