When Amplification with Weak Values Fails to Suppress Technical Noise

The application of postselection to a weak quantum measurement leads to the phenomenon of weak values. Expressed in units of the measurement strength, the displacement of a quantum coherent measuring device is ordinarily bounded by the eigenspectrum of the measured observable. Postselection can enable an interference effect that moves the average displacement far outside this range, bringing practical benefits in certain situations. Employing the Fisher-information metric, we argue that the amplified displacement offers no fundamental metrological advantage, due to the necessarily reduced probability of success. Our understanding of metrological advantage is the possibility of a lower uncertainty in the estimate of an unknown parameter with a large number of trials. We analyze a situation in which the detector is pixelated with a finite resolution and in which the detector is afflicted by random displacements: imperfections that degrade the fundamental limits of parameter estimation. Surprisingly, weak-value amplification is no more robust to them than a technique making no use of the amplification effect brought about by a final, postselected measurement.

Popular Summary

The world of quantum mechanics has never ceased to surprise. “Weak-value measurements,” discovered in 1988, are an intriguing example. Usually, when a quantum system is very weakly coupled to a measuring device, the change in the value of the measuring device is tiny. The surprise comes when the system, beginning in a definite state before the measurement, is subsequently forced into a certain unlikely state afterwards. Because of the randomness in quantum theory, many attempts are needed to achieve this. When one succeeds, something remarkable happens. There is an amplification effect: The observed change in the value of the measuring device pointer is much larger than usual. Somehow, the result of the weak measurement appears to be more powerful when the measured system makes an unlikely transition during the interaction.

Ever since this discovery, scientists have pondered its usefulness in technology as a means for signal amplification. In this paper, we show that, unfortunately, the weak-value-amplification effect cannot outperform a more straightforward approach to determining small signals.

The key theoretical tool we have used in our analysis is known as the Fisher information, a fundamental concept in statistical and information theory. It serves as the measure of the true performance of any signal-amplification scheme, including those based on weak-value measurements. Weak-value amplification requires special circumstances to operate, which carry a cost. We have found that, when such costs are taken into account, the apparent advantages of the weak-value-amplification approach disappear. The new results apply even when the experiment is limited by detector imperfections that would seem to favor amplified signals.

Our work has important implications: It clarifies the circumstances in which weak-value amplification can be a cheaper or easier technique than other methods, and it highlights the availability of powerful standard approaches to the estimation of small signals, which require less complicated experimental control and are just as robust to detector imperfections.

Figure 1

The AAV setup. A particle’s spatial wave function is weakly coupled to an internal observable $\mathbf{A}$, causing it to undergo a small lateral shift conditional on the internal state. It is then strongly postselected into the eigenstate of another observable $\mathbf{B}$ and allowed to impinge on a array of pixels with width $r$. By tuning certain parameters, one can induce larger-than-usual average displacements at the detection stage. This figure illustrates the $\text{dim}(\mathcal{H})=2$ case.

Figure 2

Numerically obtained relationships between relative Fisher information $\alpha$ and inverse resolution $R$. The different curves (blue to red; top to bottom in the main figure and right to left in the inset) correspond to different alignments that are schematized in the insets. The worst and best cases at $h=0$ and $h=0.5$, respectively, are shown in bold. Aside from misalignment effects (which become important when $R\gtrsim 3$), finite resolution does not dramatically limit parameter estimation. In fact, with only two pixels, the penalty paid is only a loss of about a third of the information, as long as good alignment is possible. The curves are equally relevant for the standard strategy or the WVA strategy.

Figure 3

Numerically obtained relationships between Fisher information and Gaussian spread ${\mathrm{\Delta }}_s$ for fixed pixel width $r_s=1$. The bold curves are for $h=0$ (lowermost) and $h=0.5$ (uppermost), with fainter curves interpolating linearly. Without pixelation, narrower Gaussian distributions are superior—the delta function offers the ultimate precision. However, the misalignment effect introduced with pixelation combines with the low effective resolution as ${\mathrm{\Delta }}_s$ is reduced, preventing the continued improvement. For ${\mathrm{\Delta }}_s\gtrsim r_s/3$, alignment ceases to be important. At any given degree of misalignment (which is described by $h$), there will be an optimum choice ${\mathrm{\Delta }}_s={\mathrm{\Delta }}_s^{*}$. The inset shows how this optimum changes with $h$.

Figure 4

The corrected Fisher information for the WVA strategy is shown in units of the standard strategy information as a contour plot in the measurement strength $G$ and the pre- and postselection angles ${\theta }_i$ and ${\theta }_f$. Two-dimensional slices through this three-dimensional plot are taken for interesting parameter combinations. The three uppermost contour plots indicate parameter combinations for which WVA can mimic the standard technique (red regions). From left to right: In the limit of a strong measurement, $G\to \infty$; when preselection and postselection are parallel, ${\theta }_i={\theta }_f$; and when preselection is into an eigenstate of the control observable, ${\theta }_i=0$. The three lowermost contour plots indicate combinations for which the WVA technique can exhibit characteristically amplified centroids. From left to right: In the limit of a weak measurement, $G\to 0$; when pre- and postselection are chosen as orthogonal, ${\theta }_f={\theta }_i+\pi /2$; and when preselection is into a state unbiased with respect to the control observable, ${\theta }_i=\pi /2$. The three-dimensional plot isosurfaces have colors corresponding to fractional values of $q{F}_g[P_{\mathrm{WVA}}]/F_g[P_{\text{std}}]$ (the color map is given in the legend); in the two-dimensional plots, the same colors shade regions between contour lines. The symmetries of the parameter space allow us to restrict ${\theta }_i\in [0,\pi ]$ with no loss in generality.

Figure 5

Contour plot of the corrected ratio of Fisher information for the weak-value-amplification strategy to Fisher information for the standard strategy, expressed as a function of the measurement strength $G$ and the angle between pre- and postselection ${\varphi }_i-{\varphi }_f$. The plot relates to imaginary weak values.