Distinguishability Measures and Ensemble Orderings

It is shown that different distinguishability measures impose different orderings on ensembles of $N$ pure quantum states. This is demonstrated using ensembles of equally-probable, linearly independent, symmetrical pure states, with the maximum probabilities of correct hypothesis testing and unambiguous state discrimination being the distinguishability measures. This finding implies that there is no absolute scale for comparing the distinguishability of any two ensembles of $N$ quantum states, and that distinguishability comparison is necessarily relative to a particular discrimination strategy.