Achieving metrological precision limits through postselection

Postselection strategies have been proposed with the aim of amplifying weak signals, which may help to overcome detection thresholds associated with technical noise in high-precision measurements. Here we use an optical setup to experimentally explore two different postselection protocols for the estimation of a small parameter: a weak-value amplification procedure and an alternative method that does not provide amplification but nonetheless is shown to be more robust for the sake of parameter estimation. Each technique leads approximately to the saturation of quantum limits for the estimation precision, expressed by the Cramér-Rao bound. For both situations, we show that parameter estimation is improved when the postselection statistics are considered together with the measurement device.

Figure 1

Experimental setup. The goal is to estimate the deflection of a mirror, represented by $g$. See the text for a complete description of the experiment.

Figure 2

Measured postselection probability as a function of the initial polarization state for strategies $|{\psi }_f\rangle ={\widehat{\sigma }}_3|{\psi }_i\rangle$ (top) and $|{\psi }_f\rangle =|{\psi }_i\rangle$ (bottom).

Figure 3

Measured meter shift as a function of the initial polarization state for strategies $|{\psi }_f\rangle ={\widehat{\sigma }}_3|{\psi }_i\rangle$ (top) and $|{\psi }_f\rangle =|{\psi }_i\rangle$ (bottom). Shown is the inferred shift of the Gaussian wave packet using the BBS technique for different initial polarization states in the Bloch sphere.

Figure 4

Estimated value of $g\mathrm{\Delta }$ from the experiment, for postselection in $|{\psi }_f\rangle ={\widehat{\sigma }}_3|{\psi }_i\rangle$. Blue circles correspond to estimates from measurements on the meter, while red squares correspond to estimates obtained via the statistics of postselection events. Error bars represent 3-$\sigma$ dispersion. Due to the mutually exclusive nature of the information obtained from meter and postselection statistics, there is no advantage in considering them together in the estimation of $g\mathrm{\Delta }$.

Figure 5

Comparison between theoretical and experimental uncertainties for estimation of the dimensionless parameter $g\mathrm{\Delta }$ as a function of the initial state $|{\psi }_i\rangle$, for the postselection $|{\psi }_f\rangle ={\sigma }_3|{\psi }_i\rangle$. The measured uncertainties are shown for estimation via the statistics of postselection events (red squares) and measurement on the meter state (blue circles). The red solid and blue dash-dotted lines show the corresponding quantum bounds, taking into account the imperfections in the interferometer.

Figure 6

Estimated value of $g\mathrm{\Delta }$ from the experiment, for postselection in $|{\psi }_f\rangle =|{\psi }_i\rangle$. Blue circles correspond to estimates from measurements on the meter, red squares correspond to estimates obtained via the statistics of postselection events, and black triangles correspond to estimates using information of both the meter and the statistics of postselection events. Error bars represent 3-$\sigma$ dispersion.

Figure 7

Comparison between theoretical and experimental uncertainties for estimation of the dimensionless parameter $g\mathrm{\Delta }$ as a function of the initial-state parameter ${\theta }_i$, for the postselection $|{\psi }_f\rangle =|{\psi }_i\rangle$. The measured uncertainties (red squares, blue circles, and black triangles) are shown for estimation via the statistics of postselection events (top), measurement on the meter state (middle), and using information of both the meter and the statistics of postselection events (bottom). The lines show the corresponding quantum bounds on precision (see the text). The scale of the bottom figure is highly amplified.