Quantum distinguishability measures: Projectors versus state maximization

The distinguishability between two quantum states can be defined in terms of their trace distance. The operational meaning of this definition involves a maximization over measurement projectors. Here we introduce an alternative definition of distinguishability which, instead of projectors, is based on maximization over normalized states (density matrices). It is shown that this procedure leads to a distance (between two states) that in contrast to the usual approach based on a 1-norm, is based on an infinite-norm. Properties such as convexity, monotonicity, and invariance under unitary transformations are fulfilled. Equivalent operational implementations based on maximization over classical probabilities and hypothesis-testing scenarios are also established. When considering the action of completely positive transformations, contractivity is only granted for unital maps. This feature allows us to introduce a measure of the quantumness of nonunital maps that can be written in terms of the proposed distinguishability measure and corresponds to the maximal possible deviation from contractivity. Particular examples sustain the main results and conclusions.