Adaptive Weak-Value Amplification Method

Since the proposal in 1988 by Aaronnov,Albert and Vaidman,the weak measurement has been widely applicated.Different from the previous quantum measurement which perform a direct measurement on the joint state,there will be an extra postselection process before performing a measurement on the meter in weak measurement.In the weak coupling of the system and the meter,the postselection process can result in the so-call weak value and thus leading to an amplification for small signal.When the pre-and-postselection are merely orthogonal,the weak value is far away from the range of engenvalues and at the same time as a complex value weak measurement shows advantages in a lot of aspects.Its potential in unknown parameter estimation is in the urgent need of kinds of research.However,the amplification for small signal of weak measurement is at the cost of the decrease of the detection probability.This causes the doubt about whether the weak measument has advantages in parameter estimation.To illustrate the problem,the Fisher information is introduced.Fisher information is a way of measuring the amount of information that an observable random variable arries about an unknown parameter of a distribution that models random variable.Thed maximum Fisher information over all measurement is the quantum Fisher information.The Cramer-Rao lower bound expressed by the Fisher information defines the precision limit in parameter estimation.gengerally,the more Fisher information is obtained,the precision for parameter estimation is higher.it has been proven that there always exists an optimal POVM measurement that can saturate the quantum Fisher information.In this paper,we discuss the Fisher information and quantum Fisher information in weak measurement.Obviously,the different weak measurement scheme will lead to a different Fisher information,and the optimal pre-and-postselection can also saturate the quantum Fisher information.By careful computation,we obtain the optimal pre-andpostselection in weak-value amplification measurement and the optimal pre-and-postselection depends on the unknown coupling parameter.We further propose an adaptive weak-value amplification scheme to approximate the optimal pre-and-postselection by adjusting the pre-andpostselection in measurement.By applying this scheme to a time-delay measurement scenario,we show by numerical simulation that the precision achieved in our scheme is several times higher than the standard WVA scheme.Our result might open a path for improving the WVA technique in a more flexible and robust way...