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arXiv:1410.0943 · Physical Review A

Weak values as interference phenomena

Justin Dressel

Published Mar 18, 2015• 127 citations
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Abstract

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.

DOI
10.1103/physreva.91.032116

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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.
Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.
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.
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.
Weak values are quantities accessed through quantum experiments involving weak measurements and postselection. It has been shown that ``anomalous'' weak values (those lying beyond the eigenvalue range of the corresponding operator) defy classical explanation in the sense of requiring contextuality [M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014)]. Here we elaborate on and extend that result in several directions. First, the original theorem requires certain perfect correlations that can never be realized in any actual experiment. Hence, we provide theorems that allow for a noise-robust experimental verification of contextuality from anomalous weak values, and compare with a recent experiment. Second, the original theorem connects the anomaly to contextuality only in the presence of a whole set of extra operational constraints. Here we clarify the debate surrounding anomalous weak values by showing that these conditions are tight: if any one of them is dropped, the anomaly can be reproduced classically. Third, whereas the original result required the real part of the weak value to be anomalous, we also give a version for any weak value with nonzero imaginary part. Finally, we show that similar results hold if the weak measurement is performed through qubit pointers, rather than the traditional continuous system. In summary, we provide inequalities for witnessing nonclassicality using experimentally realistic measurements of any anomalous weak value, and clarify what ingredients of the quantum experiment must be missing in any classical model that can reproduce the anomaly.
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.
The average result of a weak measurement of some observable A can, under postselection of the measured quantum system, exceed the largest eigenvalue of A. The nature of weak measurements, as well as the presence of postselection and hence possible contribution of measurement disturbance, has led to a long-running debate about whether or not this is surprising. Here, it is shown that such "anomalous weak values" are nonclassical in a precise sense: a sufficiently weak measurement of one constitutes a proof of contextuality. This clarifies, for example, which features must be present (and in an experiment, verified) to demonstrate an effect with no satisfying classical explanation.