Improving the Precision of Weak Measurements by Postselection Measurement

Abstract

Postselected weak measurement is a useful protocol to amplify weak physical effects. However, there has recently been controversy over whether it gives any advantage in precision. While it is now clear that retaining failed postselections can yield more Fisher information than discarding them, the advantage of postselection measurement itself still remains to be clarified. In this Letter, we address this problem by studying two widely used estimation strategies: averaging measurement results, and maximum likelihood estimation, respectively. For the first strategy, we find a surprising result that squeezed coherent states of the pointer can give postselected weak measurements a higher signal-tonoise ratio than standard ones while all standard coherent states cannot, which suggests that raising the precision of weak measurements by postselection calls for the presence of "nonclassicality" in the pointer states. For the second strategy, we show that the quantum Fisher information of postselected weak measurements is generally larger than that of standard weak measurements, even without using the failed postselection events, but the gap can be closed with a proper choice of system state. . It involves weak coupling between the system and the pointer, but the postselection on the system leads to a surprisingly counterintuitive effect: the average shift of the final pointer state can go far beyond the eigenvalue spectrum of the system observable (multiplied by the coupling constant), in sharp contrast to the projective quantum measurement. The mechanism behind this effect is the coherence between the pointer states translated by different eigenvalues of the system observable, which has an enlightening interpretation based on superoscillation [2].

Weak-value amplification: state of play

“…Further, it has been shown that a computationally trivial estimator exists for time correlated noise and outperforms the weak-valueampli cation technique [16]. If one has chosen a suboptimal initial state |i ≠ |i * , then performing an optimal postselection f * | can lead to more (corrected) Fisher information than not performing the second measurement at all (case 1) [11] -otherwise, when |i * is chosen, the Fisher information of case 4 achieves the theoretical maximum and cannot be surpassed by any pre and postselected strategy, including an optimal one.…”

“…[30][31][32]); for example with entanglement [33] or squeezing [11]. Both these e ects are known to unlock a genuine quantum advantage for certain metrological tasks, but at this point it is not clear whether combining them with weak values can indeed lead to anything greater than the sum of its parts.…”

Noise suppression in inverse weak value-based phase detection

“…For a more complete overview of both the relevant theory and experiments, see the review article by Dressel et al [21]. There has been recent theoretical work examining the use of quantum optical resources in the meter degree of freedom which shows that it is possible for a postselected measurement to achieve higher sensitivity than non-postselected measurements [22,23]. However, the overwhelming majority of experimental work to date has used classical (i.e.…”

“…However, the overwhelming majority of experimental work to date has used classical (i.e. coherent) states which can not improve over traditional schemes with postselection in ideal, shot-noise limited systems [23].…”

Protecting weak measurements against systematic errors

“…Since the Fisher information is proportional to the amount of the data and the postselection of the system discards a large portion of the measurement results, then the precision of the measurement may be lowered by the postselection. However, it was shown that the loss of Fisher information can be negligible when the initial state and the postselected state of the system are chosen properly [26,28,64,69,70], and that the signal-to-noise ratio (SNR) of postselected weak measurements can be made much higher than that of standard weak measurements by utilizing squeezed states for the probe [67].…”