The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable---or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which, when satisfied, can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak-value measurement accuracies are also briefly discussed.
The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable---or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which, when satisfied, can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak-value measurement accuracies are also briefly discussed.
In the weak value formalism of Aharonov et al., the weak value Aw of any observable A is generally a complex number. We derive a physical interpretation of its value in terms of the shift in the measurement pointer's mean position and mean momentum. In particular, we show that the mean position shift contains a term jointly proportional to the imaginary part of the weak value and the rate at which the pointer is spreading in space as it enters the measurement interaction.
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the correlations among different pointer degrees of freedom can crucially affect the way the imaginary parts of the weak values are related to the observed shifts of the mean pointer position and momentum. The particular relevance of this analysis is discussed in the case of sequential weak interactions followed by a projective measurement enabling postselection (called sequential WM) which, in the special case, reduces to the usual WM scheme (involving a single weak interaction and postseletion) modified by the effect of pointer-state correlations.
A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wave function for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wave function takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by Jozsa [R. Jozsa, Phys. Rev. A 76, 044103 (2007)]. This is even more striking in the case of sequential weak measurements [G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007)]. These are carried out by coupling several pointers at different stages of the evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wave function. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.
Exact pointer states are obtained for projection operator measurements performed upon pre-selected (PS) and upon pre- and post-selected (PPS) quantum systems. These states are used to provide simple exact expressions for both the pointer spatial probability distribution profiles and the mean values of arbitrary pointer observables associated with PS and PPS projection operator measurements that are valid for any strength of the interaction which couples a measurement pointer to the quantum system. These profiles and mean values are compared in order to identify the effects of post-selection upon projector measurement pointers. As a special case, these mean value results are applied to the weak measurement regime - yielding PS and PPS mean value expressions which are valid for any operator (projector or non-projector). Measurement sensitivities which are useful for estimating weak measurement accuracies for PS and PPS systems are also obtained and discussed.
Weak-value amplification has been successfully used to amplify the sensitivity of small detector signals, which paved the way towards new developments for precise parameter estimation methods. In general, the scheme is based on the idea that, between pre- and postselection, the measured system and the pointer are coupled in such a way that the interaction depends linearly on the coupling constant, the so-called linear regime. In this article, we propose an improvement to this technique that is based on the quadratic regime of interaction to increase the precision of such measurements. We demonstrate that by employing a purely real (imaginary) weak value in this regime, it is possible to decrease considerably the standard deviation of the final pointer observable in the position (momentum) space, when compared to the traditional linear regime, rendering a precision enhancement. By using an all-optical implementation, we obtained experimental data that confirm our theoretical predictions.
A weak measurement consists of coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so the results apply to any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{o}$ of the probe. Furthermore, a family of well-behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.