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arXiv:0706.4207 · Physical Review A

Complex weak values in quantum measurement

Richard Jozsa

Published Oct 11, 2007• 335 citations
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Abstract

In the weak value formalism of Aharonov et al., the weak value ${A}_{w}$ 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.

DOI
10.1103/physreva.76.044103

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•Mar 24, 2015•0 citations
In the weak value formalism of Aharonov et al., the weak value Aw{A}_{w}Aw​ of any observable AAA 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.
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 contribution of the detector dynamics to the weak measurement is analyzed. According to the usual theory [Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)] the outcome of a weak measurement with preselection and postselection can be expressed as the real part of a complex number: the weak value. By accounting for the Hamiltonian evolution of the detector, here we find that there is a contribution proportional to the imaginary part of the weak value to the outcome of the weak measurement. This is due to the coherence of the probe being essential for the concept of complex weak value to be meaningful. As a particular example, we consider the measurement of a spin component and find that the contribution of the imaginary part of the weak value is sizable.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
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.
In quantum mechanics, it is well known that counterfactual arguments sometimes lead to paradoxes, which implies that our classical intuition is not always reliable.However, it has been pointed out by several authors that the weak value, which was introduced by Aharonov et al. in 1988, may be useful to comprehend quantum mechanics intuitively.Another extraordinary aspect of the weak value is its amplification, which has been exploited for precision measurement.Nevertheless, the physical meaning of the weak value is still obscure due to its complexity of the value and the very fact of the amplification.In this thesis, by considering various situations of quantum interference, we attempt to provide a possible interpretation of the weak value.We find, on one hand, that the imaginary part of the weak value is relevant to the wave nature.On the other hand, in the double-slit experiment the particle nature can be seen in the real part of the position weak value.This property for the real part does not hold for more general cases such as the multiple (more than two) slit experiment.The position weak value, nevertheless, can be interpreted as the average of the classical trajectories based on a complex probability.We also investigate the possibility of knowing which path the particle takes in the multiple-slit experiment without destroying the interference fringes.We show that, because of the statistical nature of the weak value, obtaining the which-path information and the interference pattern is compatible with complementarity.Our study suggests that the weak value is a reasonable physical quantity.