Hardy's paradox

Experiments[3][4] using the technique of weak measurement[5] have studied an interaction of polarized photons, and these have demonstrated that the phenomenon does occur.

However, the consequence of these experiments is only that past events can be inferred after their occurrence as a probabilistic wave collapse.

These weak measurements are considered to be an observation themselves, and therefore part of the causation of wave collapse, making the objective results only a probabilistic function rather than a fixed reality.

However, a careful analysis of the experiment shows that Hardy's paradox only proves that a local hidden-variable theory cannot exist, as there cannot be a theory that assumes that the system meets the states of reality regardless of the interaction with the measuring apparatus.

[citation needed] This confirms that a quantum theory, to be consistent with the experiments, must be non-local (in the sense of Bell) and contextual.

The basic building block of Hardy's thought experiment are two Mach–Zehnder interferometers for quantum particles and antiparticles.

Each interferometer consists of bent paths and two beam splitters (labeled BS1 and BS2 in the accompanying diagram) and is tuned so that when operating individually, particles always exit to the same particle detector (the ones labeled c in the diagram; c is for "constructive interference" and d is for "destructive interference").

In the actual experiment the interferometers are arranged so that part of their paths overlap as shown in the diagram.

If (classically speaking) both the electron and the positron take the w paths in their respective interferometers, they will annihilate to produce two gamma rays:

Notice that a detection in both d detectors is represented by This is not orthogonal to the expression above for the state before the final beam splitters.

The situation can be analyzed in terms of two simultaneous interaction-free measurements: from the point of view of the interferometer on the left, a click at d+ implies the presence of the obstructing electron in u−.

If we assume the particles are independent (described by local hidden variables), we conclude that they can never emerge simultaneously in d+ and d−.

term arises, in fact, from the nonmaximally entangled nature of the state just before the final beam splitters.

They proposed a way that this could be observed physically by temporarily trapping the electron and the positron in the v paths in boxes and noting the effect of their mutual electrostatic attraction.

In 2009 Jeff Lundeen and Aephraim M. Steinberg published work[3] in which they set up a "Hardy's paradox" system using photons.

These then hit a beam splitter, which sends photons back to the barium borate crystal with 50% probability.

If both the 810 nm photons come back to the crystal, they are annihilated by interaction with the returning pump beam.

As predicted by Aharonov and colleagues, they found a negative value for the combination in which both photons take the outer (no-annihilation) route.