Hardy's paradox and the entanglementlike structure of forward-scattered waves

We analyze Hardy's paradox from the point of view of scattering theory. This approach has been useful for the understanding of interaction-free measurement, which is a similar setup. We calculate the forward-scattered waves generated by the beam splitters, which are replaceable in the gedanken experiment. These two-mode waves appear to have an entanglementlike structure.

Figure 1

The interferometric setup for interaction-free measurement. In the absence of absorber $A$ (say, a bomb), detector $D$ is always idle. If $A$ is there, there is a finite probability that $D$ fires, thus the interfering particle is not absorbed. (Thus the bomb is detected without exploding.)

Figure 2

The interferometric setup of Hardy's paradox. The electron and the positron may meet at point $P$ only. The beam splitters $BS$2$\pm$ are optionally replaceable; if they are replaced, the particles reach the respective detector directly. The symbols $c+,d+,c-,d-$ denote the paths in the interferometer to guide the reader's eye. The plus and minus signs in this path notation refer to the type of particle, while the letter corresponds to the detector at the direct (not reflected) end of the path.