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

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