Single-shot Distinguishability and Anti-distinguishability of Quantum Measurements (2410.10632v1)

Abstract: Among the surprising features of quantum measurements, the problem of distinguishing and anti-distinguishing general quantum measurements is fundamentally appealing. Quantum theory offers four distinct scenarios for distinguishing (and anti-distinguishing) quantum measurements - (i) probing single systems and without access to the post-measurement states, (ii) probing entangled systems and without access to the post-measurement states, (iii) probing single systems with access to the post-measurement states, and (iv) probing entangled systems with access to the post-measurement states. In these scenarios, we consider the probability of distinguishing (and anti-distinguishing) quantum measurements sampled from a given set in the single-shot regime. For some scenarios, we derive analytical expressions for the distinguishability (and anti-distinguishability) of qubit projective measurements. Notably, we demonstrate that in scenario (iii), the distinguishability of any pair of qubit projective measurements is always higher than in scenario (ii). Interestingly, for some qubit non-projective measurements, the highest distinguishability in scenario (ii) is achieved using a non-maximally entangled state. It turns out that, for any set of measurements, the distinguishability (and anti-distinguishability) in scenario (i) is always less than or equal to that in any other scenario, while it reaches its highest possible value in scenario (iv). We establish that these relations form a strict hierarchy, and there is no hierarchical relation between scenarios (ii) and (iii). Specifically, we construct pairs (and triples) of qubit measurements that are perfectly distinguishable (and anti-distinguishable) in scenario (ii) but not in scenario (iii), and vice versa. Furthermore, we identify qubit measurements that achieve perfect distinguishability (and anti-distinguishability) only in scenario (iv).