We offer the first operational interpretation of the \(\alpha \) -z relative entropies, a measure of distinguishability between two quantum states introduced by Jakšić et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the \(\alpha \) -z relative entropies for \(\alpha <1\) of the two pairs are ordered. In this setting, the \(\alpha \) and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.
