Precision Metrology Using Weak Measurements

Weak values and measurements have been proposed as a means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur with highly suppressed probabilities. This raises three vital questions in weak-measurement-based metrology. Namely, (Q1) Does postselection enhance the measurement precision? (Q2) Does weak measurement offer better precision than strong measurement? (Q3) Is it possible to beat the standard quantum limit or to achieve the Heisenberg limit with weak measurement using only classical resources? We analyze these questions for two prototypical, and generic, measurement protocols and show that while the answers to the first two questions are negative for both protocols, the answer to the last is affirmative for measurements with phase-space interactions, and negative for configuration space interactions. Our results, particularly the ability of weak measurements to perform at par with strong measurements in some cases, are instructive for the design of weak-measurement-based protocols for quantum metrology.

Figure 1

Contributions to the total information from the three constituents in the conditional-phase-rotation scenario with pre- and postselected QS state ${\psi }_i=(|-1⟩+|+1⟩)/\sqrt{2}$ and ${\psi }_f=(|-1⟩-|+1⟩)/\sqrt{2}$, and initial MA state $|\alpha ⟩$. Red curves, $F_p$. Green curves, $p_d{Q}_d$. Blue curves, $(1-p_d)Q_r$. The sum of three quantities $F_{\mathrm{tot}}$ equals $Q_j$, the total QFI of the joint system-meter state which is $4n^2+4n$ and $n=|\alpha {|}^2$.

Figure 2

Classical and quantum FIs for $g=\pi /2$. in the conditional phase-rotation scenario with ${\psi }_i=(|-1⟩+|+1⟩)/\sqrt{2}$ and ${\psi }_f=(|-1⟩-|+1⟩)/\sqrt{2}$, and initial MA state $|\alpha ⟩$. Green curve, $p_d{Q}_d$. Blue curve, $(1-p_d)Q_r$. Black curve, $Q_j=4n^2+4n$, and Brown curve, classical scaling of $4n$. $F_p$ is not shown since it is 0. The green and blue lines add up to the black line.