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arXiv:1607.08039 · New Journal of Physics

When a negative weak value −1 plays the counterpart of a probability 1

Kazuhiro Yokota, Nobuyuki Imoto

Published Dec 2, 2016• 6 citations
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Abstract

When the weak value of a projector is 1, a quantum system behaves as in that eigenstate with probability 1. By definition, however, the weak value may take an anomalous value lying outside the range of probability like −1. From the viewpoint of a physical effect, we show that such a negative weak value of −1 can be regarded as the counterpart of the ordinary value of 1. Using photons, we experimentally verify it as the symmetrical shift in polarization depending on the weak value given by pre-postselection of the path state. Unlike observation of a weak value as an ensemble average via weak measurements, the effect of a weak value is definitely confirmed in Hong–Ou–Mandel effect: the symmetrical shift corresponding to the weak value can be directly observed as the rotation angle of a half wave plate.

DOI
10.1088/1367-2630/18/12/123002

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When the weak value of a projector is 1, a quantum system behaves as in that eigenstate with probability 1. By definition, however, the weak value may take an anomalous value lying outside the range of probability like −1. From the viewpoint of a physical effect, we show that such a negative weak value of −1 can be regarded as the counterpart of the ordinary value of 1. Using photons, we experimentally verify it as the symmetrical shift in polarization depending on the weak value given by pre-postselection of the path state. Unlike observation of a weak value as an ensemble average via weak measurements, the effect of a weak value is definitely confirmed in Hong–Ou–Mandel effect: the symmetrical shift corresponding to the weak value can be directly observed as the rotation angle of a half wave plate.
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.
We propose a usage of a weak value for a quantum processing between preselection and postselection. While the weak value of a projector of 1 provides a process with certainty like the probability of 1, the weak value of −1 negates the process completely. Their mutually opposite effect is approved without a conventional 'weak' condition. In addition the quantum process is not limited to be unitary; in particular we consider a loss of photons and experimentally demonstrate the negation of the photon loss by using the negative weak value of −1 against the positive weak value of 1.
We propose a usage of a weak value for a quantum processing between preselection and postselection. While the weak value of a projector of 1 provides a process with certainty like the probability of 1, the weak value of -1 negates the process completely. Their mutually opposite effect is approved without a conventional `weak' condition. In addition the quantum process is not limited to be unitary; in particular we consider the dissipation of photons and experimentally demonstrate the negation of the dissipation by using the negative weak value of -1 against the positive weak value of 1.
Despite being very influential on both foundations and applications of quantum mechanics, weak values are still somewhat controversial. Although there are some indications that weak values are physical properties of a single quantum system, the common way weak values are presented is statistical: it is commonly believed that for measuring weak values one has to perform many weak measurements over a large ensemble of pre- and postselected particles. Other debates surround the anomalous nature of weak value and even their quantumness. To address these issues, we present some preliminary data showing that anomalous weak values can be measured using just a single detection, i.e. with no statistics. In our experiment, a single click of a detector indicates the weak value as a single photon property, which moreover lies well beyond the range of eigenvelues of the measured operator. Importantly, the uncertainty with which the weak values is measured is smaller than the difference between the weak value and the closet eigenvalue. This is the first experimental realization of robust weak measurements.
We experimentally determine weak values for a single photon's polarization, obtained via a weak measurement that employs a two-photon entangling operation, and postselection. The weak values cannot be explained by a semiclassical wave theory, due to the two-photon entanglement. We observe the variation in the size of the weak value with measurement strength, obtaining an average measurement of the S1 Stokes parameter more than an order of magnitude outside of the operator's spectrum for the smallest measurement strengths.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.