A weak values approach for testing simultaneous Einstein–Podolsky–Rosen elements of reality for non-commuting observables

In questioning the completeness of quantum mechanics, Einstein–Podolsky–Rosen (EPR) claimed that from the outcomes of local experiments performed on an entangled system, it was possible to ascribe simultaneous reality to the values of certain incompatible observables. As EPR acknowledged, the inevitable disturbance of quantum measurements prevents the precise verification of these assertions on a single system. However, the EPR elements of reality can still be tested at the ensemble level through weak measurements—which minimally disturb the measured system—by interpreting the EPR assertions as assertions about weak values that follow from the outcomes of projective measurements. Here, we report an implementation of such a test through joint weak measurements followed by post-selection on polarization-entangled photon pairs. Our results show that there is a correspondence between the obtained joint weak values and the inferred elements of reality in the polarization version of the EPR assertions.