Multiple-shot labeling of quantum observables

TL;DR

This paper investigates the limits and strategies for multiple-shot quantum labeling tasks, analyzing the success probabilities and optimal schemes for binary and non-binary observables.

Contribution

It provides the first comprehensive analysis of multiple-shot quantum labeling, including formulas for minimum-error performance and optimal schemes.

Findings

Perfect labeling impossible if single-shot impossible, regardless of number of shots.

Derived closed-form formulas for minimum-error success probabilities.

Entanglement does not improve success probability for non-binary observables.

Abstract

Quantum labeling tasks ask one to recover the missing associations between classical outcome labels and the effects forming the POVM. We study labeling in the multiple-shot regime, allowing a finite number of uses of the device and the most general tester-based strategies, including adaptivity. For binary observables, we show that if perfect labeling is impossible in a single shot, then it remains impossible with any finite number of shots. In particular, we derive the formula for minimum-error performance and highlight its ``even-odd" behavior. For non-binary observables, we derive the optimal single-shot minimum-error success probability in closed form, and show that entanglement assistance does not improve this optimum. We also provide finite-shot schemes for (perfect or partial) labeling and give illustrative examples, including the qubit trine POVM where the optimal two-shot success probability is computed explicitly.