Measuring Incompatible Observables by Exploiting Sequential Weak Values

One of the most intriguing aspects of quantum mechanics is the impossibility of measuring at the same time observables corresponding to noncommuting operators, because of quantum uncertainty. This impossibility can be partially relaxed when considering joint or sequential weak value evaluation. Indeed, weak value measurements have been a real breakthrough in the quantum measurement framework that is of the utmost interest from both a fundamental and an applicative point of view. In this Letter, we show how we realized for the first time a sequential weak value evaluation of two incompatible observables using a genuine single-photon experiment. These (sometimes anomalous) sequential weak values revealed the single-operator weak values, as well as the local correlation between them.

Figure 1

(a) Experimental setup and detection apparatus: second harmonic generator (SHG), quarter wave plate (QWP), half wave plate (HWP), polarizing beam splitter (PBS), birefringent crystal (BC). (b) Typical single data acquisition obtained with our spatial resolving detector ($32\times 32$ SPAD camera), after noise subtraction (acquisition time: 300 s). (c) The corresponding, theoretically predicted probability distribution. (d) Our SPAD camera prototype.

Figure 2

Measured weak values (data points) compared with the theoretical predictions (dashed lines) for different ${\widehat{\mathrm{\Pi }}}_{\psi }$ (i.e., for different values of $\theta$, since $|\psi ⟩=\cos \theta |H⟩+\sin \theta |V⟩$). Blue and red points and lines correspond to the evaluations of the single weak values $⟨{\widehat{\mathrm{\Pi }}}_{\psi }{⟩}_w$ and $⟨{\widehat{\mathrm{\Pi }}}_V{⟩}_w$, respectively, while purple points and line represent the evaluation of the sequential weak value $⟨{\widehat{\mathrm{\Pi }}}_{\psi }{\widehat{\mathrm{\Pi }}}_V{⟩}_w$. Uncertainty bars are evaluated on the basis of sequences of repeated measurements. The pre-selected and post-selected states are, respectively, $|{\psi }_i⟩=0.588|H⟩+0.809|V⟩$ and $|{\psi }_f⟩=|H⟩$ for plot (a), and $|{\psi }_i⟩=0.509|H⟩+0.861|V⟩$ and $|{\psi }_f⟩=-0.397|H⟩+0.918|V⟩$ for plot (b).