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

TL;DR

This paper investigates the effects of plural weak values on measurement sensitivity and systematic errors in weak value amplification, revealing anomalies and proposing methods to mitigate negative impacts.

Contribution

It uncovers the abnormal behavior caused by plural weak values and introduces three methods to reduce associated systematic errors in weak measurement protocols.

Findings

Plural weak values can decrease sensitivity under certain conditions.

Systematic errors are introduced due to device imperfections and environmental factors.

Proposed methods can mitigate negative effects of plural weak values.

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.