  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.

Quantum Physics