We offer the first operational interpretation of the $α$-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 relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the $α$-z relative entropies for $α$<1 of the two pairs are ordered. In this setting, the $α$ 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.

翻译：我们给出了$α$-z相对熵的首个操作解释，该相对熵由Jakšić等人以及Audenaert和Datta引入，用于度量两个量子态之间的可区分性。我们证明，这些相对熵出现在平面态对及其某些推广形式的大样本或催化的相对主要化条件中。具体而言，我们表明，当且仅当两个态对的所有$α<1$的$α$-z相对熵具有序关系时，这种变换才存在。在此框架下，参数$α$和$z$彼此真正独立。这些结果还给出了将一个平面态对转换为另一个的最优速率表达式。我们的方法使用了涉及预序半环及其上某些单调同态和微分的实代数技巧。