Inequalities for quantum divergences and the Audenaert-Datta conjecture

Understand

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.