Multiple-shot labeling of quantum observables

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.