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arXiv:2111.06034

The systematic error and the anomaly in the sensitivity and the probability of post-selection raised from the plural weak value

Jing-Hui Huang, Xue-Ying Duan, Guangjun Wang, Xiangyun Hu

Published Nov 11, 2021• 0 citations
arXivHTMLPDF

Abstract

Recently, weak value $\rm A_w$ derived in the pre- and post-selected weak measurement has been shown to be powerful in measuring minute physical effects. In principle, the decrease in the post-selection probability will increase the sensitivity. Besides, the sensitivity which characterizes the pointer position shift is proportional to the real part of $\rm A_w$, and the sensitivity which characterizes the pointer shift in momentum space is proportional to the imaginary part of $\rm A_w$. However, we find that the relationship between the post-selection probability and the sensitivity is true when $\rm A_w$ is a purely real number or a purely imaginary number. The plural $\rm A_w$ will lead to the abnormal behavior where the decrease in the post-selection probability reduce the sensitivity at certain pre- and post-selection. In addition, considering the device imperfections and the environmental instability in the WVA protocol, this anomaly raised from plural $\rm A_w$ will reduce the sensitivity and generate a systematic error of the measurement compared with the original scheme. Finally, three feasible methods are proposed to reduce these negative effects when the weak measurement inevitably changes into measurement with a plural weak value.

DOI
10.48550/arxiv.2111.06034

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Recently, weak value Aw\rm A_wAw​ derived in the pre- and post-selected weak measurement has been shown to be powerful in measuring minute physical effects. In principle, the decrease in the post-selection probability will increase the sensitivity. Besides, the sensitivity which characterizes the pointer position shift is proportional to the real part of Aw\rm A_wAw​, and the sensitivity which characterizes the pointer shift in momentum space is proportional to the imaginary part of Aw\rm A_wAw​. However, we find that the relationship between the post-selection probability and the sensitivity is true when Aw\rm A_wAw​ is a purely real number or a purely imaginary number. The plural Aw\rm A_wAw​ will lead to the abnormal behavior where the decrease in the post-selection probability reduce the sensitivity at certain pre- and post-selection. In addition, considering the device imperfections and the environmental instability in the WVA protocol, this anomaly raised from plural Aw\rm A_wAw​ will reduce the sensitivity and generate a systematic error of the measurement compared with the original scheme. Finally, three feasible methods are proposed to reduce these negative effects when the weak measurement inevitably changes into measurement with a plural weak value.
In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength, the imaginary one is regarded as the back-action due to the measurement with a post-selection, which has an influence on the post-selection probability. In this paper I give a case in which a real part could also appear as such a back-action in a post-selection probability on an equal footing with an imaginary one. It is also shown that both of the real and imaginary parts can be inferred by observing the probability in practice, which has an advantage that an additional system of a pointer is not needed.
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.
In the weak value formalism of Aharonov et al., the weak value Aw{A}_{w}Aw​ of any observable AAA is generally a complex number. We derive a physical interpretation of its value in terms of the shift in the measurement pointer's mean position and mean momentum. In particular, we show that the mean position shift contains a term jointly proportional to the imaginary part of the weak value and the rate at which the pointer is spreading in space as it enters the measurement interaction.
The AAV effect is the well-known phenomenon where a weak measurement followed by post-selection leads to a pointer shift proportional to the weak value of the measured observable. The effect is usually derived by considering a perturbative expansion of the time-evolution operator corresponding to the measurement. We show that the AAV effect can be instead derived given three simple conditions on the pointer states. The most important condition is that the pointer states be approximately indistinguishable. The other two conditions relate the observables used to characterize the pointer with the measured system observable. This intuitive approach does not require any model for the measurement process, but is completely consistent with the usual approach based on the von Neumann Hamiltonian. Moreover, this approach predicts and explains a new phenomenon, called weak echo, where the usual weak value behaviour can re-emerge at high interaction strengths.
Weak-value amplification has been successfully used to amplify the sensitivity of small detector signals, which paved the way towards new developments for precise parameter estimation methods. In general, the scheme is based on the idea that, between pre- and postselection, the measured system and the pointer are coupled in such a way that the interaction depends linearly on the coupling constant, the so-called linear regime. In this article, we propose an improvement to this technique that is based on the quadratic regime of interaction to increase the precision of such measurements. We demonstrate that by employing a purely real (imaginary) weak value in this regime, it is possible to decrease considerably the standard deviation of the final pointer observable in the position (momentum) space, when compared to the traditional linear regime, rendering a precision enhancement. By using an all-optical implementation, we obtained experimental data that confirm our theoretical predictions.
Abstract Weak measurement (WM) based on post-selection is a useful tool for precision metrology due to the ability of amplifying weak physical effects. In this paper, we reveal the underlying mechanism of the amplification in the post-selected WM based on the weak-value amplification (WVA) and the almost balanced weak-values amplification (ABWVA) strategies. We find that the amplification stems from the interference between the local oscillator and signal components which are respectively associated with the initial system state and the state disturbance induced by the weak coupling. The magnitude of the amplification is proportional to the interference strength which is dominated by the post-selection and the coupling strength. Furthermore, our theory can perfectly explain the amplification limit in the WVA strategy and the anomalous amplification in the ABWVA strategy.