## Abstract

Exact expressions for the mean pointer position, the mean pointer momentum and their variances are obtained for projection operator measurements performed upon ensembles of pre- and postselected (PPS) and preselected-only (PSO) quantum systems. These expressions are valid for any interaction strength which couples a measurement pointer to a quantum system, and consequently should be of general interest to both experimentalists and theoreticians. To account for the ‘collapse’ of PPS states to PSO states that occurs as interaction strength increases and to introduce the concept of ‘weak value persistence’, the exact PPS and PSO pointer theories are combined to provide a pointer theory for statistical mixtures. For the purpose of illustrating ‘weak value persistence’, mixture weights defined in terms of the Bhattacharyya coefficient are used and the statistical mixture theory is applied to mean pointer position data associated with weak value projector measurements obtained from a recent dynamical quantum non-locality detection experiment.

## 1. Introduction

The pointer of a measurement apparatus is fundamental to quantum measurement theory because the values of measured observables are determined from its properties—typically from the mean values and variances for the pointer position and momentum operators and , respectively. Understanding these properties has become increasingly important in recent years—in large part due to the renewed interest in the foundations of quantum physics, especially in the theories of weak measurements of pre- and postselected (PPS) quantum systems and weak values of quantum mechanical observables.

The use of PPS techniques for controlling and manipulating quantum systems was introduced by Schrödinger [1,2] more than 75 years ago. Since then, PPS techniques have found utility in such diverse research areas as quantum system–environment interactions (e.g. [3]), the quantum eraser (e.g. [4]) and Pancharatnam phase (e.g. [5,6]). One especially fertile area of application of PPS theory is the time symmetric reformulation of quantum mechanics developed by Aharonov *et al.* [7] and the closely related notion of the weak value of a quantum mechanical observable (e.g. [8–10]).

The weak value *A*_{w} of a quantum mechanical observable *A* is the statistical result of a standard measurement procedure performed upon a PPS ensemble of quantum systems when the interaction between the measurement pointer and each system is sufficiently weak. Unlike a standard strong measurement of *A* which significantly disturbs the measured system and yields the mean value of the associated operator as the measured value of the observable, a weak measurement of *A* performed upon a PPS system does not appreciably disturb the quantum system and yields *A*_{w} as the measured value of the observable. The peculiar nature of the virtually undisturbed quantum reality that exists between the boundaries defined by the PPS states is revealed by the eccentric characteristics of *A*_{w}, namely that *A*_{w} can be complex valued and that its real part can lie far outside the eigenvalue spectral limits of . While the interpretation of weak values remains somewhat controversial, experiments have verified several of the unusual properties predicted by weak value theory [11–15].

It is well known that projection operators, i.e. projectors, are an important part of the general mathematical formalism of quantum mechanics. These operators are also interesting because the measurement and interpretation of their weak values have played a central role in the theoretical and experimental resolutions of the ‘quantum box problem’ [16–18] and ‘Hardy's paradox’ [19–23]. More recently, projector weak values have also been used to investigate paradoxical behaviour in Mach–Zehnder interferometers (MZIs) [24] and employed in experimental observations of dynamical quantum non-locality-induced effects [25–27].

In this paper, the idempotency of projection operators is exploited in order to provide exact expressions for the mean pointer position, the mean pointer momentum, and their variances that are obtained from projector measurements performed upon ensembles of PPS quantum systems and upon ensembles of preselected-only (PSO) quantum systems. As these results are exact and valid for projector measurements of arbitrary interaction strength, they should be of general interest to both experimentalists and theoreticians.

To account for the fact that PPS states tend to ‘collapse’ into PSO states as the interaction strength increases, the exact PPS and PSO pointer theories are combined to obtain a pointer theory for statistical mixtures. For additional clarity, measurement-induced ‘collapse’ of PPS states into PSO states is discussed from an experimental perspective in the context of photon beam overlap in a MZI. An expression for determining the interaction strength after which the effect of state ‘collapse’ upon the mean pointer position becomes observable is obtained and used to define the concept of ‘*weak value persistence*’. For the purpose of illustrating ‘weak value persistence’, mixture weights defined in terms of the Bhattacharyya coefficient [28] are employed in order to compare the statistical mixture theory with mean pointer position data for weak value projector measurements obtained from a recent dynamical quantum non-locality detection experiment. Although only tangentially related to the theme of this paper, an interesting observation concerning the relationship between projector idempotency, interference and state postselection is also made and the notion of an ‘*idempotency effect eraser*’ is introduced.

The remainder of this paper is organized as follows: in §2, the exact theory for PPS projector measurement pointers is developed. In §3, the exact pointer theory for PSO projector measurements is realized and discussed. The pointer theory for PPS and PSO mixtures is obtained in §4 and compared with experimental data in §5. Closing remarks constitute the final section of this paper.

## 2. Pointer theory for pre- and postselected projector measurements

Projection operators are interesting because their idempotent property can be used to provide exact descriptions of pointers resulting from projection operator measurements. Specifically, when an instantaneous measurement is performed upon a quantum system to determine the value of a projection operator , the associated von Neumann measurement interaction operator is easily shown (using the series expansion of the measurement interaction operator and the generalized projector idempotent property , *n*≥1) to be given exactly by
2.1Here, *γ* is the measurement interaction strength and is the pointer position translation operator defined by its action upon the pointer state |*φ*〉 (it is hereafter assumed that ).

Consider the measurement at time *t* of a time-independent projection operator performed upon a PPS quantum system. If the premeasurement normalized pointer state is |*φ*〉, then—from equation (2.1)—the exact normalized state |*Φ*〉 resulting from the measurement is
2.2or
2.3where |*ψ*_{i}〉 and |*ψ*_{f}〉 are the normalized PPS states at *t*, respectively; *χ* is the Pancharatnam phase defined by [29]
*A*_{w} is the weak value of *A* at *t* given by
and
is the normalization constant. Here,
and—for any Hermitean ,

Straightforward application of equation (2.3) yields the following exact expressions for the postmeasurement mean pointer momentum and position and their variances
2.4
2.5
2.6
and
2.7where
Note that—as required—equations (2.4)–(2.7) yield , , and , respectively, when *γ*=0 and there is no measurement.

Equation (2.5) can be used to easily extend the theory to provide simple exact expressions for the maximum deflection of a pointer's mean position and the optimum *A*_{w} associated with the maximum deflection. In particular, for the typical case where *A*_{w} and 〈*q*|*φ*〉=*φ*(*q*) are real valued and |*φ*(*q*)|^{2} is symmetric about , then so that
and
where
Equation (2.5) can then be written as
2.8Setting and solving for *A*_{w} yields
2.9as the value of *A*_{w} for which is a maximum. The maximum pointer deflection for a fixed interaction strength is obtained by substituting equation (2.9) into equation (2.8)
The last two equations are valid for arbitrary *γ*≠0. That *γ*≠0 follows from the fact that *F*_{0}=2 because each of the two overlap integrals in *F*_{γ} has its maximum unit value when *γ*=0.

As has been emphasized, the above theory is exact and valid for arbitrary (‘non-collapsing’) values of the interaction strength *γ* (*γ*≠0 for maximum pointer deflection). For the special case of weak measurements where 0<|*γ*|≪1, i.e. when measurements are performed in *the weak measurement regime*, equations (2.4)–(2.7) agree precisely with the required results found in [30] for weak value measurements of any Hermitean operator —projector or not. It is *important* to observe that (i) when *A*_{w}=0,1 the exact mean pointer positions for arbitrary *γ* obtained from equation (2.5) are identical to those obtained from the associated weak value measurements performed in the weak measurement regime and (ii) *A*_{w}=0,1 are the only projector weak values for which (i) is true (these observations are fundamental to the concept of ‘weak value persistence’ defined below).

## 3. Pointer theory for preselected-only projector measurements

Consider the measurement at time *t* of a time-independent projector performed upon a system prepared in the normalized state |*ψ*〉, i.e. a PSO system. From equation (2.1), the exact normalized state |*Ψ*〉 immediately following the measurement is
3.1from which it is easily determined that
3.2
3.3
3.4
and
3.5As is the case for PPS systems, equations (3.2)–(3.5) are exact and valid for arbitrary (including ‘collapsing’) values of *γ*.

By inspection, it is seen that—unlike their counterparts for PPS measurements—the mean pointer momentum and its variance for PSO measurements do not depend upon *γ* and are unchanged by the measurement process. It is also obvious that while the mean pointer position and its variance are changed by the measurement process and depend upon *γ* for both PPS and PSO measurements, this *γ* dependence is significantly simpler for PSO measurements where it is linear in *γ* for mean position and quadratic in *γ* for its variance. This linear dependence is generally advantageous for PSO measurements because the mean value of a projector can be measured in a straightforward manner when the measurement-induced pointer translation and interaction strength are known, i.e. . Analogous linear pointer behaviour occurs for PPS measurements when the measurements are sufficiently weak and *A*_{w} is real valued or the rate of change of just prior to *t* vanishes ([30, eqn (2)]).

Using equations (3.1) and (2.2), it is readily found that the exact spatial distribution profiles of the PSO and PPS measurement pointers are 3.6and 3.7respectively. These distribution profiles are used in §4 to construct a statistical mixture pointer theory.

As a tangential point of interest, observe from this that while both of these profiles are weighted sums of the profiles for the premeasurement state |*φ*〉 and the *γ*-translated state , only the PPS profile contains a cross term and—consequently—generally exhibits interference (note that the PPS profile does not exhibit interference for the special cases *A*_{w}=0,1). Such interference does not occur for PSO measurements because of the *idempotency effect* where the idempotency of precludes the existence of an interference cross term proportional to in the PSO profile. In particular, the cross terms vanish as they contain and as factors. PPS projector measurements exhibit interference because state postselection acts as an *idempotency effect eraser* which nullifies the preventive effect of the idempotency of upon interference by replacing with *A*_{w}—thereby producing non-vanishing cross terms.

## 4. Pointer theory for pre- and postselected and preselected-only mixtures: weak value persistence

In order to account for the ‘collapse’ of PPS states into PSO states as the measurement interaction strength increases, it is assumed here that the distribution profile |〈*q*|*Ω*〉|^{2} for a PPS projector measurement pointer is the statistical mixture of PPS and PSO distributions given by
4.1where 0≤*α*,*β*≤1, *β*=1−*α*, and |〈*q*|*Ψ*〉|^{2} and |〈*q*|*Φ*〉|^{2} are given by equations (3.6) and (3.7), respectively. The associated mean pointer position is then given by
4.2where and are given by equations (2.5) and (3.3) with |*ψ*〉=|*ψ*_{i}〉, respectively. Thus, when *α*=1, *β*=0 (*α*=0, *β*=1) none (all) of the systems in the PPS ensemble are ‘collapsed’. As the measurement interaction strength increases, PPS states tend to increasingly ‘collapse’ into PSO states so that as ; the distribution |〈*q*|*Ψ*〉|^{2} becomes increasingly dominant in the mixture; and the mean pointer position is increasingly governed by .

If it is assumed that the initial pointer state is the Gaussian wavefunction
where Δ is the width of the distribution profile |〈*q*|*φ*〉|^{2}, then and . In this case, and *F*_{γ}=2 *e*^{−(γ/Δ)2} so that when *A*_{w} is real valued
4.3and equation (4.2) becomes
4.4

The concept of ‘weak value persistence’ can now be introduced by first recalling from §2 that when *A*_{w}=0,1 the exact mean pointer positions for arbitrary *γ* are identical to those for the associated weak value measurements which are obtained in the weak measurement regime and that *A*_{w}=0,1 are the only such weak values for which this is the case. Let *ϵ*>0 be a real number which characterizes the smallest change in mean pointer position that is detectable by an experimental apparatus during a PPS measurement of a projector when the change is induced by an increase in interaction strength. The interaction strength *γ*_{0} that satisfies the equality
corresponds to the smallest interaction strength for which and above which the departure of from is large enough to be observed. Upon application of equations (4.3) and (4.4), this equality assumes the form
and simplifies to
4.5when *A*_{w}=0,1.

This suggests the following definition for ‘*weak value persistence*’: suppose 0<|*γ*_{w}|≪1 is the interaction strength used to (weakly) measure either *A*_{w}=0 or *A*_{w}=1 and *γ*_{0} is the interaction strength that satisfies equation (4.5). If |*γ*_{0}|>|*γ*_{w}|, then *A*_{w}=0 or *A*_{w}=1 is said to persist over the *γ* range 0<|*γ*_{w}|≤*γ*≤|*γ*_{0}|. It is clear from equation (4.5) that the *γ* range over which *A*_{w} can persist can be relatively large for fixed *α* when is small. Thus, from an experimental perspective, ‘weak value persistence’ can be exploited to relax the rather severe weak measurement regime constraint on interaction strength when performing PPS weak value measurements of projectors with weak values of 0 or 1.

## 5. Illustrating weak value persistence using experimental data

The purpose of this section is to illustrate the notion of ‘weak value persistence’ using the data obtained from an optical experiment in which PPS measurements of the photon occupation number projector were made over a range of interaction strengths in order to detect the presence of a dynamical quantum non-locality-induced effect in a cascade of twin MZIs. In this experiment, a piezoelectrically driven computer-controlled stage produced a series of small translations of mirror M1 in the first MZI in the cascade. Each of these translations yielded a PPS measurement of in which the associated mirror displacement corresponded to the interaction strength *γ* and produced a transverse shift in the observed spatial distribution profile of the light exiting one of the ports in the output beamsplitter of the second MZI. The mean position of this profile in the laboratory reference frame served as the measurement pointer (for additional theoretical and experimental details the reader is invited to consult [25–27]).

In order to illustrate ‘weak value persistence’ using the statistical mixture theory developed in §4, let *α*=*B*, where *B*=*e*^{−D} is the Bhattacharyya coefficient which measures the amount of overlap between two distributions [28]. This choice for *α* is—in a loose sense—consistent with the experiment because the mean pointer position measurements observed at the output port of the second MZI are essentially derived from overlapping beams produced at the second beamsplitter BS2 in the first MZI (when *γ*=0 the beams traversing each arm in the first MZI completely overlap at BS2; as the mirror displacement increases—i.e. as *γ* increases—the beam overlap decreases at BS2).

When the two distributions are identical (univariate) Gaussians with their means separated by *γ*—as they ‘effectively’ are in this experiment, then
so that
where it is assumed that Δ is twice the standard deviation of the distributions. Using this and the fact that for this experiment in equations (4.4) and (4.5) gives
5.1and
For sufficiently large Δ, the last equation yields to good approximation
5.2

Now, consider the data presented in figure 1. There the vertical axis corresponds to the mean pointer position referenced to and the interaction strength is related by *γ*=−3/2*x* to a position *x* on the horizontal axes which corresponds to the translational displacement of mirror M1. The 10 data points presented in figure 1 are mean pointer positions obtained from a series of PPS measurements of projector for five equally spaced interaction strengths spanning the interval [0,500] μm. These interaction strengths clearly extended beyond the very small values required for weak value measurements (in this experiment, 0<|*γ*|≪2Δ≈300 μm is the weak measurement regime when |*A*_{w}|≤1 [12, 26, 31]). The five data points represented by circles are mean pointer positions for *A*_{w}=0 and the five data points represented by triangles are mean pointer positions for *A*_{w}=1. It should be noted that in this figure: (i) the (0,0) point corresponds to the M1 displacement for which the ‘dark’ output port of BS2 was its ‘darkest’ when *A*_{w}=0,1, whereas the (0,0) point in [26], fig. 2 was defined by the crossing of the mean pointer positions associated with the *A*_{w}=1 and the *A*_{w}=1/2 measurements (1 and 1/2 were the weak values of interest in the experiment; *A*_{w}=0 was measured to only ensure order compliance); and (ii) the *A*_{w}=0,1 mean pointer positions (effectively) coincide at (0,0) (their actual points are (0,∓5.75) μm, respectively).

Three lines are also shown on the figure. The horizontal line labelled ‘PPS theoretical no ‘collapse’ *A*_{w}=0’ is the theoretical mean pointer position obtained from equation (2.5) for the case *A*_{w}=0 and is given by . The line labelled ‘PPS theoretical no ‘collapse’ *A*_{w}=1’ is the theoretical mean pointer position obtained from equation (2.5) for the case *A*_{w}=1 and is given by . The equations for these lines are identical to the associated mean pointer position expressions given in [30] for PPS measurements with *A*_{w}=0,1 performed in the weak measurement regime. The line labelled ‘PSO theoretical 〈*A*〉=1/2’ is the mean pointer position obtained from equation (3.3) for the case and is given by . This line represents mean pointer positions for PSO systems.

Observe that the circle and triangle data points converge towards the line as their measurement interaction strengths (mirror displacements) increase. This is the expected behaviour predicted by equation (4.4) because—as interaction strength increases—more PPS systems ‘collapse’ to PSO systems so that and . In order to provide a continuous (non-discrete) representation of this ‘collapse’, equation (5.1) with *γ*=−3/2*x* was fit to the circle data points (for which *A*_{w}=0) and to the triangle data points (for which *A*_{w}=1) using Δ as the fit parameter. These curves are shown in figure 1, where the curve labelled ‘statistical mixture *A*_{w}=0’ is the *A*_{w}=0 fit (Δ=385 μm) and the curve labelled ‘statistical mixture *A*_{w}=1’ is the *A*_{w}=1 fit (Δ=500 μm).

It is important to point out that even though equation (5.1) does not provide a good fit to the *A*_{w}=0 data points, it has been included in figure 1 to highlight the approximate ‘collapse-’ related reflection symmetry (across the 1/2*γ* line) present in the experimental data. This lack of a good fit to the *A*_{w}=0 data is due in large part to the fact that these data points are significantly contaminated by noise induced by the change in position of a phase window in the apparatus that was needed to perform the *A*_{w}=0 measurements. This backfit of the phase window perturbed the precision alignment already established and used to make the *A*_{w}=1,1/2 measurements (as mentioned above, because the *A*_{w}=0 measurements were not the focus of the experiment and were obtained to merely ensure order compliance, rigorous procedures were not employed to establish a precision alignment for these measurements).

Of principal interest here is the fit of equation (5.1) to the triangle data points. The quality of this fit was somewhat degraded by the fact that the pointer states that were not pure Gaussian states (e.g. the pinhole used in the experiment produced small diffraction fringes) and by the fact that the simple mixture coefficients defined in terms of the Bhattacharyya coefficient used here do not completely capture all aspects associated with pointer ‘collapse’ (of course these factors also affected the quality of the *A*_{w}=0 data points). Nonetheless, this curve fits the first two values for corresponding to mirror displacements of 0 μm (*γ*=0 μm) and 100 μm (*γ*=−150 μm) well enough to illustrate weak value persistence by showing its existence in the *A*_{w}=1 measurement data. Using Δ=500 μm and *ϵ*=1 μm (a reasonable assumption for this experiment) in equation (5.2), it is found that |*γ*_{0}|≈100 μm which corresponds to a mirror displacement of approximately 67 μm. Inspection of the curve fit, the solid line and the first few data points in figure 1 suggests that this value is consistent with the fact that if additional *A*_{w}=1 measurements were made for mirror displacements in the range (0,67) μm the associated data points would effectively lie on the solid line—thereby exhibiting weak value persistence for measurements made with interaction strengths with absolute values in the range (0,100] μm.

## 6. Closing remarks

Exact pointer theories which are valid for arbitrary (‘non-collapsing’ for PPS systems) interaction strengths have been developed for both PPS and PSO projector measurements. Comparison of the pointer spatial distribution profiles for these two theories showed that the idempotency of the projector prevents the occurrence of interference in the profiles for PSO projector measurements, whereas state postselection nullifies this effect—thereby generally ensuring the existence of interference in PPS projector measurement profiles. Thus, state postselection acts as an *idempotency effect eraser* for projector measurements.

In order to account for the ‘collapse’ of PPS systems to PSO systems as the measurement interaction strength increases, these pointer profiles were combined to provide a statistical mixture pointer theory. This mixture theory was used to define the notion of *weak value persistence*. Using mixture weights based upon the Bhattacharyya coefficient, the statistical mixture pointer theory was compared to recent experimental mean pointer position data for *A*_{w}=0,1 weak value measurements which exhibited the effects of state ‘collapse’ with increasing interaction strength and weak value persistence was illustrated using the *A*_{w}=1 measurement data. It was noted that weak value persistence can be exploited in experiments involving weak value measurements of projectors with weak values 0,1. In particular, the rather severe weak value measurement interaction strength constraint which normally confines measurements to the weak measurement regime can be relaxed considerably for projector measurements with weak values of 0,1. As previously mentioned, such weak value projector measurements have played important roles in the experimental resolutions of certain quantum ‘paradoxes’.

## Funding statement

This work was supported in part by a grant from the Naval Surface Warfare Center Dahlgren Division's In-house Laboratory Independent Research program.

- Received September 30, 2013.
- Accepted November 20, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.