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

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.