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arXiv:1409.1535 · Physical Review Letters

Anomalous Weak Values Are Proofs of Contextuality

Matthew F. Pusey

Published Nov 12, 2014• 224 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.113.200401

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•Oct 2, 2014•0 citations
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.
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.
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.
Abstract Weak measurement offers a powerful framework for probing nonclassical features of quantum mechanics, with anomalous weak values serving as operational signatures of contextuality. While the anomalous weak value verification of quantum contextuality has been predominantly investigated in the single-photon regime and analyzed under approximation condition of infinitesimally small perturbation strength. This study releases the approximation condition and takes into account the impact of perturbation strength on the rigor of the verification. And the investigation on the verification of contextuality is extended to the multi-photon scenarios for observing the influence of the correlation between photons on the verification. Without the limitation of infinitesimally small probability of disturbance, anomalous weak values are identified as necessary for contextuality to emerge, thereby refining the criterion proposed by Pusey [ Phys. Rev. Lett. 113 200401 (2014)]. In the multi-photon scenarios, the emergence of contextuality also depends strongly on both the photon number and the photon-number distribution state. In particular, contextuality is found to be maximized when the single-photon component dominates and the second-order correlation is lower. These results highlight the critical role of photon statistics in experimental tests of contextuality via anomalous weak values.
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.
If a quantum system is prepared and later post-selected in certain states, "paradoxical" predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance required in models of these phenomena, we find that the strong measurement version of logical pre- and post-selection paradoxes actually constitute a direct manifestation of quantum contextuality. The proof hinges on under-appreciated features of the paradoxes. In particular, we show by example that it is not possible to prove contextuality without Luders-von Neumann updates for the intermediate measurements, nonorthogonal pre- and post-selection, and 0/1 probabilities for the intermediate measurements. Since one of us has recently shown that anomalous weak values are also a direct manifestation of contextuality [arXiv:1409.1535], we now know that this is true for both realizations of logical pre- and post-selection paradoxes.