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arXiv:1109.4183 · Physical Review A

Full counting statistics of weak-value measurement

Antonio Di Lorenzo

Published Mar 5, 2012• 49 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.032106

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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 o\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{o}o\ifmmode^\elseˆ\fi 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.
A measurement consists in coupling a system to a probe and reading the output of the probe to gather information about the system. The weaker the coupling, the smaller the back-action on the system, but also the less information conveyed. If the system undergoes a second measurement, the statistics of the first output can be conditioned on the value of the second one. This procedure is known as postselection. A postselected weak measurement of an observable can give a large average output of the probe when the postselected state is nearly orthogonal to the initial state of the system. This large value is an interference effect in the readout of the probe, which is initially in a coherent superposition of readout states (also known as pointer states). Usually, the weak interaction between system and probe is considered instantaneous, so that the dynamics of the probe can be neglected. However, for a weak measurement in solid-state devices, an interaction of finite duration is likely needed. Here, we show how this finite duration generates a contribution of the dynamical phase to the readout statistics. Furthermore, we derive interpolation formulas that are able to describe the statistics of the weak measurement for the whole range of pre- and postselected states. Phase-space averages appear in the expansion, suggesting an interpretation in terms of non-positive probabilities. Decoherence in the probe is also accounted for and it is pointed out the existence of a regime of intermediate coupling strength in which coherent oscillations can be observed in the probability of the readout.
In a weak measurement, the average output ⟨p⟩\langle p\rangle⟨p⟩ of a probe that measures an observable $\Hat{A}$ of a quantum system undergoing both a preparation in a state ρi\rho_iρi​ and a postselection in a state EfE_fEf​ is, to a good approximation, a function of the weak value $A_w=\Tr[E_f\Hat{A}\rho_i]/\Tr[E_f\rho_i]$, a complex number. For a fixed coupling, when the overlap $\Tr[E_f\rho_i]$ is very small, AwA_wAw​ diverges, but ⟨p⟩\langle p\rangle⟨p⟩ stays finite, often tending to zero for symmetry reasons. This paper answers the question: which are the extremal values of the output as a function of the weak value? We derive equations for the optimal values of AwA_wAw​, and provide the solutions. The results are independent of the dimensionality of the system.
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.
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 weak value, the average result of a weak measurement, has proven useful for probing quantum and classical systems. Examples include amplifying small signals, investigating quantum paradoxes, and elucidating fundamental quantum phenomena such as geometric phase. A key characteristic of the weak value is that it can be complex, in contrast to a standard expectation value. However, typically only either the real or imaginary component of the weak value is determined in a given experimental setup. Weak measurements can be used to, in a sense, simultaneously measure noncommuting observables. This principle was used in the direct measurement of the quantum wave function. However, the wave function's real and imaginary components, given by a weak value, are determined in different setups or on separate ensembles of systems, putting the procedure's directness in question. To address these issues, we introduce and experimentally demonstrate a general method to simultaneously read out both components of the weak value in a single experimental apparatus. In particular, we directly measure the polarization state of an ensemble of photons using weak measurement. With our method, each photon contributes to both the real and imaginary parts of the weak-value average. On a fundamental level, this suggests that the full complex weak value is a characteristic of each photon measured.
The variance of an observable in a preselected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and postselected systems, we formulate a complex-valued counterpart of the variance called ``weak variance.'' In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and postselected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and postselected systems.