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arXiv:1410.5221

Conditions for Anomalous Weak Value

Arun Kumar Pati, Junde Wu

Published Oct 20, 2014• 5 citations
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Abstract

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.

DOI
10.48550/arxiv.1410.5221

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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 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.
A weak value of an observable is studied for a quantum system which is placed under the influence of an environment, where a quantum system irreversibly evolves from a pre-selected state to a post-selected state. A general expression for a weak value influenced by an environment is provided. For a Markovian environment, the weak value is calculated in terms of the predictive and retrodictive density matrices, or by means of the quantum regression theorem. For a non-Markovian environment, a weak value is examined by making use of exactly solvable models. It is found that although the anomalous property is significantly suppressed by a Markovian environment, it can survive a non-Markovian environment.
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.
Weak measurement has been shown to play important roles in the investigation of both fundamental and practical problems. Anomalous weak values are generally believed to be observed only when post-selection is performed, i.e, only a particular subset of the data is considered. Here, we experimentally demonstrated an anomalous weak value can be obtained without discarding any data by performing a sequential weak measurement on a single-qubit system. By controlling the blazing density of the hologram on a spatial light modulator, the measurement strength can be conveniently controlled. Such an anomalous phenomenon disappears when the measurement strength becomes strong. Moreover, we find that the anomalous weak value can not be observed without post-selection when the sequential measurement is performed on each of the components of a two-qubit system, which confirms that the observed anomalous weak value is based on sequential weak measurement of two noncommutative operators.