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arXiv:quant-ph/0503225 · Physical Review A

Quantum averages of weak values

Yakir Aharonov, Alonso Botero

Published Nov 23, 2005• 111 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.72.052111

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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.
We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit. As such, we address the controversy in the literature regarding the theoretical consistency of the quantum weak value by providing a more general theoretical framework and giving several examples of how that framework relates to existing experimental and theoretical results.
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.
A recent Letter in Physical Review Letters, "Contextual Values of Observables in Quantum Measurements", by J. Dressel, S. Agarwal, and A. N. Jordan (abbreviated DAJ below), introduces the concept of "contextual values" and claims that they lead to "a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit". However, they do not define "minimal disturbance limit". The present paper is in part the saga of my search for a definition of "minimal disturbance limit" under which this claim could be proved. The search finally ended in what is probably a definitive counterexample to the claim.
Information about the internal properties of a system can only be obtained through interactions of the system with an external meter. However, such interactions generally result in entanglement between the system and the meter, making it difficult to trace the measurement result back to a specific value of the physical property in the system. It is therefore possible that the outcomes of quantum measurements depend in a nontrivial manner on the dynamics of the measurement interaction, possibly providing a physical explanation for the role of measurement contexts in quantum mechanics. Here we show that the effects of the measurement interaction on the meter can be described entirely in terms of the quantum coherent system dynamics associated with the back-action on the system. For sufficiently small back-action uncertainties, the physical property of the system is described by a weak value obtained from the Hamilton-Jacobi equation of the back-action dynamics. At higher measurement resolutions, the observed values are determined by quantum interferences between different amounts of back-action. Eigenvalues emerge when the quantum interferences between different back-actions correspond to a Fourier transform in the back-action parameter. We conclude that the values of physical properties obtained in quantum measurements originate from the quantum coherent properties of the back-action dynamics generated by that physical property during an interaction. Measurement outcomes represent elements of the dynamics and cannot be explained by measurement independent elements of reality.
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.
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.