Single-shot Distinguishability and Anti-distinguishability of Quantum Measurements

TL;DR

This paper investigates the fundamental problem of distinguishing and antidistinguishing quantum measurements in various scenarios, providing analytical results for qubit measurements and exploring the effects of entanglement and measurement access.

Contribution

It offers the first analytical expressions for measurement distinguishability in different scenarios and reveals the complex hierarchy and non-hierarchical relations among these scenarios.

Findings

Distinguishability in scenario (iii) exceeds that in scenario (ii) for qubit projective measurements.

Certain non-projective measurements achieve optimal distinguishability with non-maximally entangled states.

Maximum distinguishability occurs in scenario (iv), with no hierarchy between scenarios (ii) and (iii).

Abstract

Among the surprising features of quantum measurements, the problem of distinguishing and antidistinguishing general quantum measurements is fundamentally appealing. Unlike classical systems, quantum theory offers entangled states and peculiar state update rule of the post-measurement state, which gives rise to four distinct scenarios: (i) probing single systems and without access to the Post-measurement States (PMS), (ii) probing entangled systems and without access to the PMS, (iii) probing single systems with access to the PMS, and (iv) probing entangled systems with access to the PMS. We study the probability of distinguishing (and antidistinguishing) quantum measurements sampled from a given set in the single-shot regime. For some scenarios, we provide the analytical expressions of distinguishability (and antidistinguishability) for qubit projective measurements. We show that the distinguishability of any pair of qubit projective measurements in scenario (iii) is always greater than its value in scenario (ii). Interestingly, certain pairs of non-projective qubit measurements achieve optimal distinguishability in scenario (ii) with a non-maximally entangled state. In general, for any set of measurements, distinguishability (and antidistinguishability) in scenario (i) never exceeds that in any other scenario, while it reaches its highest possible value in scenario (iv). We establish that there is no hierarchical relation between scenarios (ii) and (iii). In particular, we introduce different variants of the well-known `trine' qubit measurement to construct pairs (and triples) of qubit quantum measurements such that they are perfectly distinguishable (and antidistinguishable) in scenario (ii) but not in scenario (iii), and vice versa. Additionally, we present qubit measurements that are perfectly distinguishable (and antidistinguishable) in scenario (iv) but not in any other scenarios.