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Research
Cite this article: Parks AD, Spence SE, Gray
JE. 2014 Exact pointer theories for von
Neumann projector measurements of pre- and
postselected and preselected-only quantum
systems: statistical mixtures and weak value
persistence. Proc. R. Soc. A 470: 20130651.
http://dx.doi.org/10.1098/rspa.2013.0651
Received: 30 September 2013
Accepted: 20 November 2013
Subject Areas:
quantum physics
Keywords:
weak values, weak measurement, quantum
measurement, projectors
Author for correspondence:
A. D. Parks
e-mail: [email protected]
Exact pointer theories for
von Neumann projector
measurements of pre- and
postselected and
preselected-only quantum
systems: statistical mixtures
and weak value persistence
A. D. Parks, S. E. Spence and J. E. Gray
Electromagnetic and Sensor Systems Department, Naval Surface
Warfare Center, Dahlgren, VA 22448, USA
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
2013 The Author(s) Published by the Royal Society. All rights reserved.
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2
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rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
for the pointer position and momentum operators q̂ and p̂, 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 Aw 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 Aw 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 Aw , namely that Aw can be complex valued and that
its real part can lie far outside the eigenvalue spectral limits of Â. 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
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in §4 and compared with experimental data in §5. Closing remarks constitute the final section of
this paper.
e−(i/h̄)γ Âp̂ is easily shown (using the series expansion of the measurement interaction operator
and the generalized projector idempotent property Ân = Â, n ≥ 1) to be given exactly by
e−(i/h̄)γ Âp̂ = 1̂ − Â + ÂŜ.
(2.1)
Here, γ is the measurement interaction strength and Ŝ ≡ e−(i/h̄)γ p̂ is the pointer position
translation operator defined by its action q|Ŝ|ϕ ≡ ϕ(q − γ ) upon the pointer state |ϕ (it is
hereafter assumed that [q̂, Â] = 0 = [p̂, Â]).
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
|Φ =
1
1
ψf |e−(i/h̄)γ Âp̂ |ψi |ϕ = ψf |(1̂ − Â + ÂŜ)|ψi |ϕ
N
N
(2.2)
1 iχ
e (1 − Aw + Aw Ŝ)|ϕ,
N
(2.3)
or
|Φ =
where |ψi and |ψf are the normalized PPS states at t, respectively; χ is the Pancharatnam phase
defined by [29]
ψf | ψi ;
eiχ ≡
|ψf | ψi |
Aw is the weak value of A at t given by
Aw =
ψf |Â|ψi ψf | ψi and
N=
, ψf | ψi = 0;
a + J(1̂)
is the normalization constant. Here,
a = 1 − 2 Re Aw + 2 |Aw |2
and—for any Hermitean x̂,
J(x̂) = Aw (1 − A∗w )ϕ|x̂Ŝ|ϕ + A∗w (1 − Aw )ϕ|Ŝ† x̂|ϕ.
Straightforward application of equation (2.3) yields the following exact expressions for the
postmeasurement mean pointer momentum and position and their variances
1
Φ|p̂|Φ =
M(p̂),
(2.4)
N2
1
[M(q̂) + γ |Aw |2 ],
(2.5)
Φ|q̂|Φ =
N2
1
V(p̂)
(2.6)
2Φ p =
N4
1
[V(q̂) + Q(q̂)],
(2.7)
and
2Φ q =
N4
...................................................
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 Â, the associated von Neumann measurement interaction operator
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
2. Pointer theory for pre- and postselected projector measurements
3
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where
4
and
Q(x̂) = 2(J(1̂)ϕ|x̂|ϕ − J(x̂))γ |Aw |2 + (a + J(1̂))γ 2 |Aw |2 − γ 2 |Aw |4 .
Note that—as required—equations (2.4)–(2.7) yield Φ|p̂|Φ = ϕ|p̂|ϕ, Φ|q̂|Φ = ϕ|q̂|ϕ,
2Φ p = 2ϕ p and 2Φ q = 2ϕ q, 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 Aw associated with the
maximum deflection. In particular, for the typical case where Aw and q|ϕ = ϕ(q) are real valued
and |ϕ(q)|2 is symmetric about ϕ|q̂|ϕ = 0, then [ϕ|q̂Ŝ|ϕ + ϕ|Ŝ† q̂|ϕ] = 0 so that
M(q̂) = J(q̂) = Aw (1 − Aw )[ϕ|q̂Ŝ|ϕ + ϕ|Ŝ† q̂|ϕ] = 0
and
N2 = a + Aw (1 − Aw )Fγ ,
where
Fγ = [ϕ|Ŝ|ϕ + ϕ|Ŝ† |ϕ].
Equation (2.5) can then be written as
Φ|q̂|Φ =
γ A2w
1 − (2 − Fγ )Aw + (2 − Fγ )A2w
.
(2.8)
Setting ∂Φ|q̂|Φ/∂Aw = 0 and solving for Aw yields
Aw =
2
2 − Fγ
(2.9)
as the value of Aw for which Φ|q̂|Φ is a maximum. The maximum pointer deflection for a fixed
interaction strength is obtained by substituting equation (2.9) into equation (2.8)
Φ|q̂|Φmax =
4γ
.
(2 − Fγ )(2 + Fγ )
The last two equations are valid for arbitrary γ = 0. That γ = 0 follows from the fact that F0 = 2
because each of the two overlap integrals in Fγ 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 O(γ ) results found in
[30] for weak value measurements of any Hermitean operator —projector or not. It is important
to observe that (i) when Aw = 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) Aw = 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
...................................................
V(x̂) = a2 2ϕ x + a(J(x̂2 ) + J(1̂)ϕ|x̂2 |ϕ − 2ϕ|x̂|ϕJ(x̂)) − J2 (x̂) + J(1̂)J(x̂2 )
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
M(x̂) = aϕ|x̂|ϕ + J(x̂),
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state |Ψ immediately following the measurement is
from which it is easily determined that
Ψ |p̂|Ψ = ϕ|p̂|ϕ,
(3.2)
Ψ |q̂|Ψ = ϕ|q̂|ϕ + γ ψ|Â|ψ,
(3.3)
and
2Ψ p = 2ϕ p
(3.4)
2Ψ q = 2ϕ q + γ 2 2ψ A.
(3.5)
As 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. ψ|Â|ψ = (Ψ |q̂|Ψ − ϕ|q̂|ϕ)/γ . Analogous linear pointer
behaviour occurs for PPS measurements when the measurements are sufficiently weak and Aw is
real valued or the rate of change of 2ϕ q 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
|q|Ψ |2 = (1 − ψ|Â|ψ)|q|ϕ|2 + ψ|Â|ψ|q|Ŝ|ϕ|2
and
2
|q|Φ| =
1
N2
|1 − Aw |2 |q|ϕ|2 + |Aw |2 |q|Ŝ|ϕ|2
,
+ 2 Re [Aw (1 − A∗w )q|ϕ∗ q|Ŝ|ϕ]
(3.6)
(3.7)
respectively. 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 Ŝ|ϕ,
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 Aw = 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
Req|ϕ∗ q|Ŝ|ϕ in the PSO profile. In particular, the cross terms vanish as they contain ψ|Â(1̂ −
Â)|ψ = ψ|(Â − Â2 )|ψ = ψ|(Â − Â)|ψ = 0 and ψ|(1̂ − Â)Â|ψ = 0 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 Â|ψ
with Aw —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
...................................................
(3.1)
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|Ψ = e−(i/h̄)γ Âp̂ |ψ|ϕ = (1̂ − Â + ÂŜ)|ψ|ϕ
5
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measurement pointer is the statistical mixture of PPS and PSO distributions given by
|q|Ω|2 = α|q|Φ|2 + β|q|Ψ |2 ,
6
(4.1)
where is the width of the distribution profile |q|ϕ|2 , then ϕ|q̂|ϕ = 0 = [ϕ|q̂Ŝ|ϕ + ϕ|Ŝ† q̂|ϕ]
2
2
and ϕ|Ŝ|ϕ = ϕ|Ŝ† |ϕ = e−(γ /) . In this case, M(q̂) = 0 and Fγ = 2 e−(γ /) so that when Aw is real
valued
γ A2w
(4.3)
Φ|q̂|Φ =
2
a + 2Aw (1 − Aw ) e−(γ /)
and equation (4.2) becomes
Ω|q̂|Ω = γ
α
A2w
a + 2Aw (1 − Aw ) e−(γ /)
2
+ βψi |Â|ψi .
(4.4)
The concept of ‘weak value persistence’ can now be introduced by first recalling from §2
that when Aw = 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 Aw = 0, 1 are the only such weak values for which this is the case. Let > 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 γ0 that satisfies the equality
|Ω|q̂|Ω − Φ|q̂|Φ| = corresponds to the smallest interaction strength for which and above which the departure of
Ω|q̂|Ω from Φ|q̂|Φ is large enough to be observed. Upon application of equations (4.3) and
(4.4), this equality assumes the form
A2w
=
γ0 (1 − α) ψi |Â|ψi −
2
−(γ
/)
0
a + 2Aw (1 − Aw ) e
and simplifies to
|γ0 (1 − α)[ψi |Â|ψi − Aw ]| = ,
(4.5)
when Aw = 0, 1.
This suggests the following definition for ‘weak value persistence’: suppose 0 < |γw | 1 is the
interaction strength used to (weakly) measure either Aw = 0 or Aw = 1 and γ0 is the interaction
strength that satisfies equation (4.5). If |γ0 | > |γw |, then Aw = 0 or Aw = 1 is said to persist over
the γ range 0 < |γw | ≤ γ ≤ |γ0 |. It is clear from equation (4.5) that the γ range over which Aw
can persist can be relatively large for fixed α when |ψi |Â|ψi − Aw | 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.
...................................................
−∞
where Φ|q̂|Φ and Ψ |q̂|Ψ are given by equations (2.5) and (3.3) with |ψ = |ψi , 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 β → 1 as α → 0; the distribution |q|Ψ |2 becomes increasingly dominant in the
mixture; and the mean pointer position Ω|q̂|Ω is increasingly governed by Ψ |q̂|Ψ .
If it is assumed that the initial pointer state is the Gaussian wavefunction
1/4
4
2
e−2(q/) ,
q|ϕ =
π 2
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
where 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
+∞
q|q|Ω|2 dq = αΦ|q̂|Φ + βΨ |q̂|Ψ ,
(4.2)
Ω|q̂|Ω =
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5. Illustrating weak value persistence using experimental data
7
α = e−(1/2)(γ /) ,
2
where it is assumed that is twice the standard deviation of the distributions. Using this and the
fact that for this experiment ψi |Â|ψi = 1/2 in equations (4.4) and (4.5) gives
Ω|q̂|Ω =
and
γ
2
[1 + (−1)Aw +1 e−(1/2)(γ /) ]
2
(5.1)
γ
2 0
[1 − e−(1/2)(γ0 /) ] = .
2
For sufficiently large , the last equation yields to good approximation
|γ0 | ≈ [42 ]1/3 ,
Aw = 0, 1.
(5.2)
Now, consider the data presented in figure 1. There the vertical axis corresponds to the mean
pointer position referenced to ϕ|q̂|ϕ = 0 and the interaction strength is related by γ = −3/2x 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 | 2 ≈ 300 µm is the weak
measurement regime when |Aw | ≤ 1 [12,26,31]). The five data points represented by circles are
mean pointer positions for Aw = 0 and the five data points represented by triangles are mean
pointer positions for Aw = 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 Aw = 0, 1,
whereas the (0, 0) point in [26, fig. 2] was defined by the crossing of the mean pointer positions
associated with the Aw = 1 and the Aw = 1/2 measurements (1 and 1/2 were the weak values
of interest in the experiment; Aw = 0 was measured to only ensure order compliance); and
...................................................
so that
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
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
1 γ 2
D=
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100
8
PPS theoretical no ‘collapse’ Aw= 0
...................................................
rspa.royalsocietypublishing.org Proc. R. Soc. A 470: 20130651
0
statistical
mixture Aw = 0
–100
pointer displacement (µm)
–200
PSO theoretical
<A> = 1/2
–300
PPS theoretical no
‘collapse’ Aw = 1
–400
statistical
mixture Aw = 1
–500
–600
–700
–800
–900
0
100
200
300
400
500
600
mirror displacement (µm)
Figure 1. Variation of mean pointer position as interaction strength increases (data points from [26]).
(ii) the Aw = 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’ Aw = 0’ is the theoretical mean pointer position obtained from equation (2.5) for the
case Aw = 0 and is given by Φ|q̂|Φ = 0. The line labelled ‘PPS theoretical no ‘collapse’ Aw = 1’
is the theoretical mean pointer position obtained from equation (2.5) for the case Aw = 1 and is
given by Φ|q̂|Φ = γ = −3/2x. The equations for these lines are identical to the associated O(1)
mean pointer position expressions given in [30] for PPS measurements with Aw = 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 A ≡ ψi |Â|ψi = 1/2 and is given by
Ψ |q̂|Ψ = 1/2γ = −3/4x. This line represents mean pointer positions for PSO systems.
Observe that the circle and triangle data points converge towards the Ψ |q̂|Ψ = 1/2γ 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 α → 0 and β → 1. In order to provide a continuous
(non-discrete) representation of this ‘collapse’, equation (5.1) with γ = −3/2x was fit to the circle
data points (for which Aw = 0) and to the triangle data points (for which Aw = 1) using as the
fit parameter. These curves are shown in figure 1, where the curve labelled ‘statistical mixture
Aw = 0’ is the Aw = 0 fit ( = 385 µm) and the curve labelled ‘statistical mixture Aw = 1’ is the
Aw = 1 fit ( = 500 µm).
It is important to point out that even though equation (5.1) does not provide a good fit to
the Aw = 0 data points, it has been included in figure 1 to highlight the approximate ‘collapse-’
related reflection symmetry (across the 1/2γ line) present in the experimental data. This lack of a
good fit to the Aw = 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 Aw = 0 measurements. This backfit of the phase window perturbed
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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 Aw = 0, 1 weak value
measurements which exhibited the effects of state ‘collapse’ with increasing interaction strength
and weak value persistence was illustrated using the Aw = 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.
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6. Closing remarks
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the precision alignment already established and used to make the Aw = 1, 1/2 measurements (as
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Of principal interest here is the fit of equation (5.1) to the triangle data points. The quality of
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