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Weak values and neoclassical realism

Josiah Sinclair, David Spierings, Aharon Brodutch, Aephraim M. Steinberg

Published Aug 29, 2018• 2 citations
OpenAlex

Abstract

Constructing an ontology for quantum theory is challenging, in part due to unavoidable measurement back-action. The Aharonov-Albert-Vaidman weak measurement formalism provides a method to predict measurement results (weak values) in a regime where back-action is negligible. The weak value appears analogous to a classical conditional mean and in fact, has a number of features that further suggest it may be interpreted as being related to some underlying ontological model. However, the ontology appears bizarre since the weak values are complex and unbounded. Here, we study weak values in the context of a recent quantum optical experiment involving two-photon interactions. The results of the experiment are reinterpreted within a 'stochastic optics' model of light. The model is based on standard (Maxwell) electromagnetic theory, supplemented by stochastic fluctuations of the electromagnetic fields. We show that the conditional means of the electric field intensities correspond to the experimentally observed weak values. This is a provocative result, as it suggests that at least within this experiment, the weak value gives us information about the average of an ontological quantity (the intensity). We study the breakdown of the stochastic optics model, which occurs outside the experimentally probed regime, and in particular in the limit where the weak value predicts 'anomalous' results.

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Constructing an ontology for quantum theory is challenging, in part due to unavoidable measurement back-action. The Aharonov-Albert-Vaidman weak measurement formalism provides a method to predict measurement results (weak values) in a regime where back-action is negligible. The weak value appears analogous to a classical conditional mean and in fact, has a number of features that further suggest it may be interpreted as being related to some underlying ontological model. However, the ontology appears bizarre since the weak values are complex and unbounded. Here, we study weak values in the context of a recent quantum optical experiment involving two-photon interactions. The results of the experiment are reinterpreted within a 'stochastic optics' model of light. The model is based on standard (Maxwell) electromagnetic theory, supplemented by stochastic fluctuations of the electromagnetic fields. We show that the conditional means of the electric field intensities correspond to the experimentally observed weak values. This is a provocative result, as it suggests that at least within this experiment, the weak value gives us information about the average of an ontological quantity (the intensity). We study the breakdown of the stochastic optics model, which occurs outside the experimentally probed regime, and in particular in the limit where the weak value predicts 'anomalous' results.
Weak values as introduced by Aharonov, Albert, and Vaidman (AAV) are ensemble-average values for the results of weak measurements. They are interesting when the ensemble is preselected on a particular initial state and postselected on a particular final measurement result. It is shown that weak values arise naturally in quantum optics, as weak measurements occur whenever an open system is monitored (as by a photodetector). The quantum-trajectory theory is used to derive a generalization of AAV's formula to include (a) mixed initial conditions, (b) nonunitary evolution, (c) a generalized (nonprojective) final measurement, and (d) a non-back-action-evading weak measurement. This theory is applied to the recent cavity-QED experiment demonstrating wave particle duality [G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, Phys. Rev. Lett. 85, 3149 (2000)]. It is shown that the ``fractional-order'' correlation function measured in that experiment can be recast as a weak value in a form as simple as that introduced by AAV.
Abstract Is it possible that a measurement of a spin component of a spin-1/2 particle yields the value 100? In 1988 Aharonov, Albert and Vaidman argued that upon pre- and postselection of particular spin states, weakening the coupling of a standard measurement procedure ensures this paradoxical result1. This theoretical prediction, called weak value, was realised in numerous experiments2–9, but its meaning remains very controversial10–19, since its “anomalous” nature, i.e., the possibility to exceed the eigenvalue spectrum, as well as its “quantumness” are debated20–22. We address these questions by presenting the first experiment measuring anomalous weak values with just a single click, without the need for statistical averaging. The measurement uncertainty is significantly smaller than the gap between the measured weak value and the nearest eigenvalue. Beyond clarifying the meaning of weak values, demonstrating their non-statistical, single-particle nature, this result represents a breakthrough in understanding the foundations of quantum measurement, showing unprecedented measurement capability for further applications of weak values to quantum photonics.
Recently, Kocsis et al (2011 Science 332 1170) reported the observation of 'average trajectories of single photons' in a two-slit interference experiment. This was possible by using the quantum weak-measurement method, which implies averaging over many events, i.e. in fact, a multi-photon limit of classical linear optics. We give a classical-optics interpretation of this experiment and other related problems. It appears that weak measurements of the local momentum of photons made by Kocsis et al represent measurements of the Poynting vector in an optical field. We consider both the real and imaginary parts of the local momentum and show that their measurements have been realized in classical optics using small-probe particles. We also examine the appearance of 'anomalous' values of the local momentum: either negative (backflow) or exceeding the wavenumber (superluminal propagation). These features appear to be closely related to vortices and evanescent waves. Finally, we revisit a number of older works and find examples of photon trajectories and anomalous-momentum measurements in various optical experiments.
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al., hereafter referred to as AAV. Their analysis led to the controversial result that the weak values for an observable can lie well outside its associated eigenvalue spectrum. We have performed an experiment similar to one proposed by Knight et al. and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity defined by Duck et al. It is shown that these properties are produced by interference phenomena that result from the loss of ‘welcher Weg’ information. We have also identified an intrinsic ‘weak energy’ which appears in the equation of motion for the weak value of an observable and have shown that it can be non–vanishing only for systems which are both preselected and postselected. It is noted that such a ‘weak spin–rotation coupling energy’ can be associated with our experiment and that it is manifested in our experimental data. We suggest that preselection and postselection measurement techniques could be useful for the creation and study of physical phenomena that might be otherwise unobservable.
We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a definite mechanical effect on a measuring probe specifically designed to minimize the back reaction on the measured system. We then present an interesting framework for general measurement conditions (where the back reaction on the system may not be negligible) in which the measurement outcomes can still be interpreted as quantum averages of weak values. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.
Weak values arise experimentally as conditioned averages of weak (noisy) observable measurements that minimally disturb an initial quantum state, and also as dynamical variables for reduced quantum state evolution even in the absence of measurement. These averages can exceed the eigenvalue range of the observable ostensibly being estimated, which has prompted considerable debate regarding their interpretation. Classical conditioned averages of noisy signals only show such anomalies if the quantity being measured is also disturbed prior to conditioning. This fact has recently been rediscovered, along with the question whether anomalous weak values are merely classical disturbance effects. Here we carefully review the role of the weak value as both a conditioned observable estimation and a dynamical variable, and clarify why classical disturbance models will be insufficient to explain the weak value unless they can also simulate other quantum interference phenomena.