Lundeen Steinberg claim to resolve Hardy paradox

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Lucien Hardy's QM paradox goes back to 1992 or 1993 when he was a grad student at Durham UK.

Lundeen and Steinberg's paper was published in Physical Review Letters January 16.
They confirm Hardy's paradox, and offer a theoretical explanation or "resolution" involving negative probabilities
==exerpt from Physorg==
"For nearly a century, the widespread interpretation of quantum mechanics suggests that everything is uncertain until it is observed, and that observation inevitably alters reality," says Professor Steinberg. "However, in the 1990s, a technique known as 'interaction-free measurement' seemed to promise the ability to 'see without looking,' as a Scientific American article put it at the time. But when Lucien Hardy proposed that one could never reliably make inferences about past events which hadn't been directly observed, a paradox emerged which suggested that whenever one attempted to reason about the past in this way they would be led into error."

Over the course of nearly two years of work, Steinberg and then-student Jeff Lundeen, now a research associate at the National Research Council of Canada, built a complicated quantum optical experiment and developed new theoretical tools. In essence, they combined Hardy's Paradox with a new theory known as weak measurement proposed by Tel Aviv University physicist Yakir Aharonov, showing that in one sense, one can indeed talk about the past, resolving the paradox. Weak measurement is a tool whereby the presence of a detector is less than the level of uncertainty around what is being measured, so that there is an imperceptible impact on the experiment. "We found that all of the seemingly paradoxical conclusions in Hardy's Paradox can, in fact, be experimentally verified," says Steinberg, "but that the use of weak measurement removes the contradiction."

The preprint of the Lundeen Steinberg article is available on arxiv. 4 pages, posted in late October 2008.
==sample quotes from pages 1 and 3 of preprint==

"Hardy’s Paradox is a contradiction between classical reasoning and the outcome of standard measurements on an electron E and positron P in a pair of Mach-Zehnder interferometers(seeFig. 1). Each interferometer is first aligned so that the incoming particle always leaves through the same exit port, termed the “bright” port B (the other is the “dark”port D). The interferometers are then arranged so that one arm (the ”Inner” arm I) from each interferometer overlaps at Y. It is assumed that if the electron and positron simultaneously enter this arm they will collide and annihilate with 100% probability. This makes the interferometers “Interaction-Free Measurements” (IFM)[6]: that is, a click at the dark port indicates the interference was disturbed by an object located in one of the interferometer arms, without the interfering particle itself having traversed that arm.

Therefore, in Hardy’s Paradox a click at the dark port of the electron (positron) indicates that the positron (electron) was in the Inner arm. Consider if one were to detect both particles at the dark ports. As IFMs, these results would indicate the particles were simultaneously in the Inner arms and, therefore should have annihilated. But this is in contradiction to the fact that they were actually detected at the dark ports. Paradoxically, one does indeed observe simultaneous clicks at the dark ports[7], just as quantum mechanics predicts..."

"...Examining the table reveals that the single-particle weak measurements are consistent with the clicks at each dark port; as the IFM results imply, the weakly measured occupations of each of the Inner arms are close to one and those of each of the Outer arms are close to zero. The weak measurements indicate that, at least when considered individually, the photons were in the Inner arms.

However, if we instead examine the joint occupation of the two Inner arms, it appears that the two photons are only simultaneously present roughly one quarter of the time. This demonstrates that, as we expect, the particles are not in the inner arms together.

So far, we seem to have confirmed both of the premises of Hardy’s Paradox: to wit, that when DP and DE fire, N(IP) and N (IE) are close to one (since the IFMs indicate the presence of the particles in Y) – but that N(IP & IE ) is close to zero (since when both particles are in Y, they annihilate and should not be detected).

This is odd because in classical logic, N(IP & IE) must be ≥ N(IP) + N(IE) − 1; this inequality is violated by our results.

Although N(IE) is 93% and N(IP ) is 92%, the data in Table 1 suggest that when E is in the Inner path, P is not, and vice versa; hence the large values for N(IE & OP) = 64% and N(OE & IP) = 72%. The fact that the sum of these two seemingly disjoint joint-occupation probabilities exceeds 1 is the contradiction with classical logic..."

IP means photon P goes by the inner path, OP means it goes by the outer. Similarly IE and OE. So N(IP & IE) is the percentage of trials where both photons went by the inner path. The two photons are called P and E because they play the roles that were played by the positron and the electron in Lucien Hardy's original thought experiment.
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Thanks for sharing.


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There is no such thing as "Hardy paradox" to begin with, as Hardy never presented his work as a "paradox". He presented it as a proof of quantum nonlocality, similar to the Bell's proof but without inequalities.

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