Information amplification via postselection: A parameter-estimation perspective

It is known that weak measurement can significantly amplify the mean of measurement results, sometimes out of the range limited in usual quantum measurement. This fact, as actively demonstrated recently in both theory and experiment, implies the possibility to estimate a very small parameter using the weak measurement technique. But does the weak measurement really bring about the increase of “information” for parameter estimation? This paper clarifies that, in a general situation, the answer is NO; more precisely, the weak measurement cannot further decrease the lower bound of the estimation error, i.e., the so-called Cramér-Rao bound, which is proportional to the inverse of the quantum Fisher information.

Figure 1

Procedure of the postselection.

Figure 2

Fisher informations versus the parameter $\theta$. The blue region represents the set of curves of $I\left({\widehat{\rho }}_{\mathrm{ps}}^{\mathcal{K}}\right)$, while the yellow dashed line indicates $I\left({\widehat{\rho }}_{\mathrm{int}}^{\mathcal{H}\otimes \mathcal{K}}\right)$. The red dashed-dotted and green dotted lines show the classical Fisher information $I_{\mathrm{c}}\left(f_{\mathrm{ps}}\left(x\right)\right)$ with $c=1$ and $c=-1$, respectively.

Figure 3

(a) The success probability $\mathrm{Pr}\left(f\right)$ and (b) Fisher information normalized by $\mathrm{Pr}\left(f\right)$, as functions of $\theta /\sigma$. The regions represent the set of curves of $\mathrm{Pr}\left(f\right)$ and $\mathrm{Pr}\left(f\right)I\left({\widehat{\rho }}_{\mathrm{ps}}^{\mathcal{K}}\right)$ generated with various values of $(t_1,t_2,s_1-s_2)$.