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arXiv:1410.8067

Classical correlation alone supplies the anomaly to weak values

Christopher Ferrie, Joshua Combes

Published Oct 29, 2014• 1 citation
arXivHTMLPDF

Abstract

The question of what is genuinely quantum about weak values is only ever going to elicit strongly subjective opinions---it is not a scientific question. Good questions, when comparing theories, are operational---they deal with the unquestionable outcomes of experiment. We give the anomalous shift of weak values an objective meaning through a generalization to an operational definition of anomalous post-selected averages. We show the presence of these averages necessitate correlations in every model giving rise to them---quantum or classical. Characterizing such correlations shows that they are ubiquitous. We present the simplest classical example without the need of disturbance realizing these generalized anomalous weak values.

DOI
10.48550/arxiv.1410.8067

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•Oct 2, 2014•0 citations
The question of what is genuinely quantum about weak values is only ever going to elicit strongly subjective opinions---it is not a scientific question. Good questions, when comparing theories, are operational---they deal with the unquestionable outcomes of experiment. We give the anomalous shift of weak values an objective meaning through a generalization to an operational definition of anomalous post-selected averages. We show the presence of these averages necessitate correlations in every model giving rise to them---quantum or classical. Characterizing such correlations shows that they are ubiquitous. We present the simplest classical example without the need of disturbance realizing these generalized anomalous weak values.
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.
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.
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.
The quantum mechanical weak value Aw=⟨ϕ∣A∣ψ⟩/⟨ϕ∣ψ⟩A_w=\left\langle ϕ|A|ψ\right \rangle / \left\langle ϕ| ψ\right\rangleAw​=⟨ϕ∣A∣ψ⟩/⟨ϕ∣ψ⟩ of an observable AAA is a measurable quantity associated with an observable AAA and pre- and post-selected states ∣ψ⟩,∣ϕ⟩\vertψ\rangle, \vert ϕ\rangle∣ψ⟩,∣ϕ⟩. Much has been discussed about the meaning and metrological uses of anomalous weak values, lying outside of the range of eigenvalues of AAA. We present a simple proof that anomalous weak values require that the (possibly mixed) pre- and post- selection states have coherence in the eigenbasis of AAA. We also present conditions under which anomalous AwA_wAw​ are witnesses of generalized contextuality, dispensing with the operational weak measurement set-up.
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.