Linear optics implementation of weak values in Hardy’s paradox

We propose an experimental setup for the implementation of weak measurements in the context of the gedanken experiment known as Hardy’s paradox. As Aharonov et al. [Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, Phys. Lett. A301, 130 (2002)] showed, these weak values form a language with which the paradox can be resolved. Our analysis shows that this language is indeed consistent and experimentally testable. It also reveals exactly how a combination of weak values can give rise to an apparently paradoxical result.

Figure 1

The original setup of Hardy’s paradox. It comprises two interleaving Mach-Zehnder interferometers, traversed simultaneously by a positron and an electron, respectively, and arranged in such a way that two arms overlap. As a result the electron and positron cannot both travel through the overlapping arm of the respective interferometer without annihilating. The removal of any one or both the final beam splitters in this setup leads to paradoxical results which serve as a proof of quantum mechanical nonlocality.

Figure 2

The photon, incident with polarization angle $\theta$, is split into its horizontal $\left(H\right)$ and vertical $\left(V\right)$ polarization components at a polarizing beam splitter $\left(A\right)$. These then reach the second polarizing beam splitter $\left(B\right)$ at different times with $\Delta t=ϵ∕c$. Thus the arrival time has become entangled with the polarization which, together with the postselection achieved by the third polarizing beam splitter $\left(C\right)$, corresponds to a weak measurement. However, this is only the case if the uncertainty in arrival time is similar in size to $ϵ$, the separation of the wave packets. In other words, $ϵ\simeq c∕4\pi \Delta \nu$, where $\Delta \nu$ is the uncertainty in frequency. The basis of beam splitter $C$ is rotated by an angle $\varphi$.

Figure 3

Two pairs of maximally entangled photons are created simultaneously in the two sources marked with $S$. Of each pair, one photon is sent through a polarizing beam splitter $\left(A\right)$ and then reunited with its counterpart from the other pair, in a conventional beam splitter $\left(B\right)$. A detector $\left(D^{\prime }\right)$ is placed behind one of the exits of this beam splitter. If it fires, the photons to the right of the sources are projected into the desired state. (Note that in an actual experiment the two pairs would most likely originate from one LBO crystal.) The other two photons each traverse a copy of the weak measurement setup described in Fig. 2, with $\theta =0$ and $\varphi =-\pi ∕4$. The detectors $D$ both click in the case of a successful bipartite postselection. For the single-particle measurements only the weak measurement setup in the path of the photon to be measured is used, and the other is omitted.