Quantum sensors based on weak-value amplification cannot overcome decoherence

Sensors that harness exclusively quantum phenomena (such as entanglement) can achieve superior performance compared to those employing only classical principles. Recently, a technique based on postselected, weakly performed measurements has emerged as a method of overcoming technical noise in the detection and estimation of small interaction parameters, particularly in optical systems. The question of which other types of noise may be combated remains open. We here analyze whether the effect can overcome decoherence in a typical field-sensing scenario. Benchmarking a weak, postselected measurement strategy against a strong, direct strategy, we conclude that no advantage is achievable, and that even a small amount of decoherence proves catastrophic to the weak-value amplification technique.

Figure 1

A schematic for two different approaches to parameter estimation using spins; the view is into the equatorial plane of the Bloch sphere. A direct strategy relies only on interrogations of the system spin, which picks up a field-dependent phase $\exp \left(it\delta B\right)$ over time. A WVA strategy makes use of an ancillary spin in the hope of gaining an advantage. The latter technique involves coupling the first spin with a second meter spin with variable strength, and then interrogating the meter spin only if the system spin is successfully postselected into a certain final state. Such a postselection can be achieved by the use of projective measurements in an appropriate basis given by $\theta$, and is known to lead to a larger than expected deflection of the meter spin if the coupling $G$ is weak. Note that while $\delta B$ is an unknown quantity of interest, $\theta ,\Theta ,$ and $G$ are tunable by the experimenter in the WVA strategy (their meanings are explained in the main text).

Figure 2

Contour lines of four important quantities are plotted against measurement strength $G$ and the angle ($\theta -t\delta B$) between pre- and postselection. Upper left: $H_{\mathrm{wva}}$:$H_{\mathrm{d}}$ uncorrected for the low postselection probability. White regions show a ratio higher than 1 and hence the apparent superiority of the WVA scheme over a direct scheme. Upper right: the inverse ratio; white regions now show where the WVA scheme is inferior. Lower left: the postselection success probability $q$. Lower right: the corrected ratio $q{H}_{\mathrm{wva}}$:$H_{\mathrm{d}}$; note that there are no longer white regions and the weak strategy is never better. The asterisk denotes a choice of $(G,\theta )$ that is generalized to include dephasing noise in Fig. 4.

Figure 3

Ratio of information available in the WVA strategy and in its direct counterpart, as a function of the measurement strength $G$. The different curves correspond to various values of the attenuation function $\Xi \left(t\right)$. The postselection is fixed to $\theta =t\delta B+\pi$. When $G=1$, one should match the direct strategy; only the postselection probability $q=1/2$ implies by symmetry that half of the information is thrown away (see Appendix pp2).

Figure 4

Quantum Fisher information (in units of ${\mathrm{T}}^{-2}$) for the two competing strategies is plotted as a function of time $t$ and the dephasing rate $\Gamma$ (for $\Xi =e^{-\Gamma t}$). The upper green surface corresponds to the direct strategy $H_{\mathrm{d}}$ and the lower purple surface corresponds to $q{H}_{\mathrm{wva}}$, the corrected weak-value amplified strategy. The postselection and measurement strength are fixed to $\theta =t\delta B+\pi ,G=0.02$, respectively, corresponding to the asterisk in Fig. 2. Even a moderate amount of dephasing has a catastrophic effect on the weak-value scheme.

Figure 5

Quantum Fisher information for a weak-value amplified sensing strategy (a) when the postselection is “successful,” (b) when the postselection is “unsuccessful,” and (c) when both are considered together; the plots shown correspond to the decoherence-free case.