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arXiv:quant-ph/0408153 · Physical Review A

Linear optics implementation of weak values in Hardy’s paradox

Sebastian E. Ahnert, Mike C. Payne

Published Oct 20, 2004• 23 citations
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Abstract

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.

DOI
10.1103/physreva.70.042102

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Comment on “Linear optics implementation of weak values in Hardy’s paradox”

Jeff S. Lundeen, Kevin J. Resch, Aephraim M. Steinberg

A recent experimental proposal [S.E. Ahnert and M.C. Payne, Phys. Rev. A 70, 042102 (2004)] outlines a method to measure the weak value predictions of Aharonov in Hardy's paradox. This proposal contains flaws in the state preparation method and the procedure for carrying out the requisite weak measurements. We identify previously published solutions to some of the flaws.

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Reply to “Comment on ‘Linear optics implementation of weak values in Hardy’s paradox’ ”

Sebastian E. Ahnert, Mike C. Payne

The comment by Lundeen et al. contains two criticisms of our proposal. While we agree that the state-preparation procedure could be replaced by a simpler setup as proposed by the authors of the comment, we do not agree with the authors on their second, and more important point regarding two-particle weak measurements. We believe this to be the result of a misunderstanding of our original paper [S. E. Ahnert and M. C. Payne, Phys. Rev. A 70, 042102 (2004)].

arXiv:quant-ph/0505027•Physical Review A•Jul 11, 2005•5 citations

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Measuring the weak value in an optical experiment

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The first experimental realization of a measurement of the weak value of a variable, a concept recently introduced by Aharonov, Albert and Vaidman, is presented. Weak measurements (measurements of a weak value) address the situation in which the separation of the eigenvalues caused by a weakly interacting measurement device is small compared to the width of the distribution of the individual eigenvalues. By appropriate choice of the pre- and post-selected state it is possible that the overlapping eigenvalues will interfere producing a value outside the range of eigenvalues. We demonstrate that the weak value is a practical method of amplifying and resolving the separation between overlapping eigenvalues in an optical experiment proposed by Duck, Stevenson and Sudarshan. In this experiment a birefringent crystal spatially separates two linear polarization components of a Gaussian laser beam by a distance much smaller than the beam waist.

•Jan 1, 1991•0 citations
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.
It has been proposed that the ability to perform joint weak measurements on postselected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here we experimentally perform weak measurements of joint (i.e., nonlocal) observables. In an implementation of Hardy's paradox, we weakly measure the locations of two photons, the subject of the conflicting statements behind the paradox. Remarkably, the resulting weak probabilities verify all of these statements but, at the same time, resolve the paradox.
The first experimental realization of a measurement of the weak value of a variable, a concept recently introduced by Aharonov, Albert and Vaidman, is presented. Weak measurements (measurements of a weak value) address the situation in which the separation of the eigenvalues caused by a weakly interacting measurement device is small compared to the width of the distribution of the individual eigenvalues. By appropriate choice of the pre- and post-selected state it is possible that the overlapping eigenvalues will interfere producing a value outside the range of eigenvalues. We demonstrate that the weak value is a practical method of amplifying and resolving the separation between overlapping eigenvalues in an optical experiment proposed by Duck, Stevenson and Sudarshan. In this experiment a birefringent crystal spatially separates two linear polarization components of a Gaussian laser beam by a distance much smaller than the beam waist.