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Journal of Physics A Mathematical and Theoretical

Typical weak and superweak values

Michael Berry, Pragya Shukla

Published Aug 12, 2010• 37 citations
Publisher

Abstract

Weak values, resulting from the action of an operator on a preselected state when measured after postselection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be ‘superweak’. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized lorentzian form, and the ‘superweak probablility’, of weak values outside the spectrum, can be as large as 1–1/ = 0.293.... By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately gaussian for many eigenvalues, is not universal.

DOI
10.1088/1751-8113/43/35/354024

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Weak values, resulting from the action of an operator on a preselected state when measured after postselection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be ‘superweak’. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized lorentzian form, and the ‘superweak probablility’, of weak values outside the spectrum, can be as large as 1–1/ = 0.293.... By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately gaussian for many eigenvalues, is not universal.
Weak value is a physical property of a quantum system which manifests itself through a weak measurement using different preand post-selected ensembles of the system. The weak values of an operator may differ significantly from its eigenvalues and can even lie outside the spectrum if it is bound: they can be ‘superweak’. The latter, originating due to a coherent superposition of waves, may appear as a ‘supershift’ on the measuring device. This property has potential application in the amplification and detection of extremely weak signals.
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.
Abstract Weak values characterize a quantum system in the period of time between preparation and measurement and may lie outside the eigenvalue spectrum of the measured operator. The probability of such “superweak" values for random quantum states has been calculated and applied to Klein–Gordon and Dirac waves, where the maximal probability for superluminal propagation was shown to be 1/2. In a recent paper, a different definition for the velocity of a relativistic quantum particle was proposed in terms of a ratio of two weak values. In this paper, we find the probability distribution of such ratios. With the new definition, the superluminal probability of photons is found to be bounded between 1−1/21-1/\sqrt{2}1−1/2​ and 1/21/\sqrt{2}1/2​ , while for general eigenvalue distributions the superluminal probability can take any value between 0 and 1.
In Y. Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988)], it was shown that, in some weak measurements with postselection schemes, it is possible to obtain inferred mean values outside the range of the eigenvalues of the measured observable. In our work, we are going to show that this effect can be associated with the presence of an asymptotically vanishing but nonzero amount of system-detector quantum correlations. These correlations are also related to the presence of a finite (nonzero) amount of coherence of the initial states in the basis of the measured observable.
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.
The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable A^\hat {A}A^ with post-selection 〈F| may be out of the eigenvalue range of A^\hat {A}A^ . This is because the weak value of A^\hat {A}A^ with the post-selection is, in general, not the expectation value of A^\hat {A}A^ , but the expectation value of A^∣F⟩⟨F∣\hat {A}| F\rangle \langle F|A^∣F⟩⟨F∣ boosted by the post-selection.