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Self-testing represents the strongest form of certification of a quantumsystem. Here we investigate theoretically and experimentally the question ofself-testing non-projective quantum measurements. That is, how can one certify,from observed data only, that an uncharacterised measurement device implementsa desired non-projective positive-operator-valued-measure (POVM). We consider aprepare-and-measure scenario with a bound on the Hilbert space dimension, whichwe argue is natural for this problem since any measurement can be madeprojective by artificially increasing the Hilbert space dimension. We developmethods for (i) robustly self-testing extremal qubit POVMs (which featureeither three or four outcomes), and (ii) certify that an uncharacterised qubitmeasurement is non-projective, or even a genuine four-outcome POVM. Our methodsare robust to noise and thus applicable in practice, as we demonstrate in aphotonic experiment. Specifically, we show that our experimental data impliesthat the implemented measurements are very close to certain ideal three andfour outcome qubit POVMs, and hence non-projective. In the latter case, thedata certifies a genuine four-outcome qubit POVM. Our results open interestingperspective for strong `black-box' certification of quantum devices.

Self-testing non-projective quantum measurements in prepare-and-measure experiments