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Hardy's Paradox

I want to discuss one further argument against local realism from Lucien Hardy in another well known set of publications, that were submitted in 1992 and 1993. In two papers titled "Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories" and "Nonlocality for two particles without inequalities for almost all entangled states". These papers present what is more commonly known as "Hardy's paradox".

These must be the nail in the coffin for any reasonable person who wants to still defend local realism. The experiment shows that the predictions of quantum mechanics differ from that of any local realist and local hidden variable theory.

The experiment involves using a stream of positions and electrons both sent through separate beam splitters on the left and the right. The wave function of each particle "splits" along the w path and the v path. These are then rejoined at the second beam splitter in phase so that if there's no interaction between the electrons and the positrons then an interference pattern is detected at c (constructive interference). 

However if you set up the experiment so that the elections on the right interfere with the positrons on the left, the electrons and positrons will annihilate producing photons


So that only particles in the v path reach the second beam splitter. These waves are not coherent with anything so they meet at detectors c and d with equal probability. We can therefore assume that any detection at either d detector means that there was an anihilation event between the two beams at point P. Whenever a detection is made at d the corresponding particle is at u so that



However if we analyse this experiment from the perspective of a local hidden variable theory, wherein particles are independent there can be no detection at d because they would have to travel though the u path where here is an annihilation event. Local hidden variables have a problem then, because these experiments have been performed and there is a detection at d with a Pr(d) =1/16. 

There's a Pr(e+) and Pr(e-) entering both w beams of 1/2 each, so 1/4 and then a further 1/4 probability of each reaching either d detector. So both taken together give a probability of 1/16.

Some more context is probably necessary, if one thinks of quantum theory as being local and as maintaining realism then there is a frame of reference in which a detection can be made at the d detector for the positron, while the corresponding particle in the electron beam has not reached the beam splitter. Similarly you can can have a frame of reference in the opposite direction in which the reverse is also true, where the electron has reached the detector but the positron has not yet reached the beam splitter. Both of these claims cannot be correct, if you compare both frames of reference the particles must have come through either w path and anihilated. So we've arrived at Hardy's paradox.

One possibility is that the particles detection at d is not independent of the second particle in u and there was some influence that occurred faster-than-light or another is that perhaps the measuring device does not register a property the particle had before detection and so there was no definite measurement outcome. The experiment appears to confirm the long held belief that any realist or hidden variable theory must deny locality in order to be consistent with the predictions of quantum mechanics. 

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