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arXiv:2303.08700

A simple proof that anomalous weak values require coherence

Rafael Wagner, Ernesto F. Galvão

Published Mar 15, 2023• 1 citation
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Abstract

The quantum mechanical weak value $A_w=\left\langle ϕ|A|ψ\right \rangle / \left\langle ϕ| ψ\right\rangle$ of an observable $A$ is a measurable quantity associated with an observable $A$ and pre- and post-selected states $\vertψ\rangle, \vert ϕ\rangle$. Much has been discussed about the meaning and metrological uses of anomalous weak values, lying outside of the range of eigenvalues of $A$. 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 $A$. We also present conditions under which anomalous $A_w$ are witnesses of generalized contextuality, dispensing with the operational weak measurement set-up.

DOI
10.48550/arxiv.2303.08700

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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 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 show that the weak value of any observable in pre- and post-selected states can be expressed as the sum of the average of the observable in the pre-selected state and an anomalous part. We argue that at a fundamental level the anomalous nature of the weak values arises due to the interference between the post-selected state and another quantum state which is orthogonal to the pre-selected state. This provides a necessary and sufficient condition for the anomalous nature of the weak value of a quantum observable. Furthermore, we prove that for two non-commuting observables the product of their anomalous parts cannot be arbitrarily large.
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.
In Y. Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988)], it was shown that, in some weak measurements with postselection schemes, it is possible to obtain inferred mean values outside the range of the eigenvalues of the measured observable. In our work, we are going to show that this effect can be associated with the presence of an asymptotically vanishing but nonzero amount of system-detector quantum correlations. These correlations are also related to the presence of a finite (nonzero) amount of coherence of the initial states in the basis of the measured observable.