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arXiv:1312.3036 · Progress of Theoretical and Experimental Physics

Weak value as an indicator of back-action

Riuji Mochizuki

Published Apr 1, 2014• 2 citations
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Abstract

In this study we critically examine some important papers on weak measurement and weak values. We find some insufficiency and mistakes in these papers, and we demonstrate that the real parts of weak values provide the back-action to the post-selection, which is caused by weak measurement. Two examples, a counterfactual statement of Hardy's paradox and experiments that determine the average trajectory of photons passing through double slits, are investigated from our point of view.

DOI
10.1093/ptep/ptu045

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In this study we critically examine some important papers on weak measurement and weak values. We find some insufficiency and mistakes in these papers, and we demonstrate that the real parts of weak values provide the back-action to the post-selection, which is caused by weak measurement. Two examples, a counterfactual statement of Hardy's paradox and experiments that determine the average trajectory of photons passing through double slits, are investigated from our point of view.
In this study we critically examine some important papers on weak measurement and weak values. We find some insufficiency and mistakes in these papers, and we demonstrate that the real parts of weak values pro-vide the back-action to the post-selection, which is caused by weak mea-surement. Two examples, a counterfactual statement of Hardy’s paradox and experiments that determine the average trajectory of photons passing through double slits, are investigated from our view point. PACS numbers: 03.65.Ta, 03.65.Ud, 03.65.Ca 1
In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength, the imaginary one is regarded as the back-action due to the measurement with a post-selection, which has an influence on the post-selection probability. In this paper I give a case in which a real part could also appear as such a back-action in a post-selection probability on an equal footing with an imaginary one. It is also shown that both of the real and imaginary parts can be inferred by observing the probability in practice, which has an advantage that an additional system of a pointer is not needed.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
Weak values are conventionally extracted in the limit of vanishing measurement strength. At finite strength, the corresponding post-selected conditional averages generally acquire nonlinear corrections and need not retain their weak value interpretation. Here we identify a class of measurement protocols for which this expectation fails: the conditional value reconstructed from the measuring probe is exactly independent of the probe strength and therefore coincides with the real part of the weak value. We first analyze this phenomenon in the measurement--disturbance protocol introduced by Lund and Wiseman. We then establish conditions under which this invariance persists, showing that it is determined by the interplay between the probe interaction, a subsequent measurement interaction with an ancillary apparatus, and the final post-selection. These results demonstrate that weak coupling, although generally sufficient for accessing weak values, is not always necessary.
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.