Protecting weak measurements against systematic errors

In this work, we consider the systematic error of quantum metrology by weak measurements under decoherence. We derive the systematic error of maximum likelihood estimation in general to the first-order approximation of a small deviation in the probability distribution and study the robustness of standard weak measurement and postselected weak measurements against systematic errors. We show that, with a large weak value, the systematic error of a postselected weak measurement when the probe undergoes decoherence can be significantly lower than that of a standard weak measurement. This indicates another advantage of weak-value amplification in improving the performance of parameter estimation. We illustrate the results by an exact numerical simulation of decoherence arising from a bosonic mode and compare it to the first-order analytical result we obtain.

Figure 1

Ratio between the systematic error with and without postselection of the system for postselected states with varying $\delta$.

Figure 2

Ratio between the systematic error with and without postselection of the system as a function of the interaction parameter $g$.

Figure 3

Ratio between the systematic error with and without postselection of the system as a function of the probe decoherence strength ${\epsilon }_D$. ${\epsilon }_D=0$ is removed from the plot, since the systematic errors for both postselected and standard weak measurements are zero at that point, leading to an ill-defined ratio between them.