-
Colloquium: Understanding Quantum Weak Values: Basics and
Applications
Justin Dressel,1, 2 Mehul Malik,3, 4 Filippo M. Miatto,5 Andrew
N. Jordan,1 and Robert W. Boyd3, 51Department of Physics and
Astronomy and Center for Coherence and Quantum Optics,University of
Rochester,Rochester, New York 14627,USA2Department of Electrical
Engineering,University of California,Riverside, California
92521,USA3The Institute of Optics,University of
Rochester,Rochester, New York 14627,USA4Institute for Quantum
Optics and Quantum Information (IQOQI),Austrian Academy of
Sciences,Boltzmanngasse 3, A-1090 Vienna,Austria5Department of
Physics,University of Ottawa,Ottawa, Ontario,Canada
(Dated: April 2, 2014)
Since its introduction 25 years ago, the quantum weak value has
gradually transitionedfrom a theoretical curiosity to a practical
laboratory tool. While its utility is apparent inthe recent
explosion of weak value experiments, its interpretation has
historically been asubject of confusion. Here, a pragmatic
introduction to the weak value in terms of mea-surable quantities
is presented, along with an explanation of how it can be
determinedin the laboratory. Further, its application to three
distinct experimental techniques isreviewed. First, as a large
interaction parameter it can amplify small signals abovetechnical
background noise. Second, as a measurable complex value it enables
noveltechniques for direct quantum state and geometric phase
determination. Third, as aconditioned average of generalized
observable eigenvalues it provides a measurable win-dow into
nonclassical features of quantum mechanics. In this selective
review, a singleexperimental configuration is used to discuss and
clarify each of these applications.
CONTENTS
I. Introduction 1
II. What is a weak value? 2
III. How does one measure a weak value? 3
IV. How can weak values be useful? 5A. Weak value amplification
5B. Measurable complex value 7C. Conditioned average 7
V. Conclusions 9
Acknowledgments 9
References 9
I. INTRODUCTION
Derived in 1988 by Aharonov, Albert, and Vaidman(Aharonov et
al., 1988; Duck et al., 1989; Ritchie et al.,1991) as a “new kind
of value for a quantum vari-able” that appears when averaging
preselected and post-
selected weak measurements, the quantum weak valuehas had an
extensive and colorful theoretical history(Aharonov et al., 2010;
Aharonov and Vaidman, 2008;Kofman et al., 2012; Shikano, 2012).
Recently, however,the weak value has stepped into a more public
spotlightdue to three types of experimental applications. It is
ouraim in this brief and selective review to clarify these
threepragmatic roles of the weak value in experiments.
First, in its role as an evolution parameter, a largeweak value
can help to amplify a detector signal and en-able the sensitive
estimation of unknown small evolutionparameters, such as beam
deflection (Dixon et al., 2009;Hogan et al., 2011; Hosten and
Kwiat, 2008; Jayaswalet al., 2014; Pfeifer and Fischer, 2011;
Starling et al.,2009; Turner et al., 2011; Zhou et al., 2013,
2012),frequency shifts (Starling et al., 2010a), phase
shifts(Starling et al., 2010b), angular shifts (Magana-Loaizaet
al., 2013), temporal shifts (Brunner and Simon, 2010;Strübi and
Bruder, 2013), velocity shifts (Viza et al.,2013), and even
temperature shifts (Egan and Stone,2012). Paradigmatic optical
experiments that have used
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this technique include the measurement of 1Å resolutionbeam
displacements due to the quantum spin Hall effectof light “without
the need for vibration or air-fluctuationisolation” (Hosten and
Kwiat, 2008), an angular mirrorrotation of 400frad due to linear
piezo motion of 14fmusing only 63µW of power postselected from
3.5mW to-tal beam power (Dixon et al., 2009), and a
frequencysensitivity of 129kHz/
√Hz obtained with 85µW of power
postselected from 2mW total beam power (Starling et al.,2010a).
All these results were obtained in modest table-top laboratory
conditions, which was possible since thetechnique amplifies the
signal above certain types of tech-nical noise backgrounds (e.g.,
electronic 1/f noise or vi-bration noise) (Feizpour et al., 2011;
Jordan et al., 2014;Knee and Gauger, 2014; Starling et al.,
2009).
Second, in its role as a complex number whose real andimaginary
parts can both be measured, the weak valuehas encouraged new
methods for the direct measurementof quantum states (Lundeen and
Bamber, 2012, 2014;Lundeen et al., 2011; Malik et al., 2014;
Salvail et al.,2013) and geometric phases (Kobayashi et al., 2010,
2011;Sjöqvist, 2006). These methods express abstract theoret-ical
quantities such as a quantum state in terms of com-plex weak
values, which can then be measured experi-mentally. Notably, the
real and imaginary componentsof a quantum state in a particular
basis can be directlydetermined with minimal postprocessing using
this tech-nique.
Third, in its role as a conditioned average of gener-alized
observable eigenvalues, the real part of the weakvalue has provided
a measurable window into nonclassi-cal features of quantum
mechanics. Conditioned averagesoutside the normal eigenvalue range
have been linkedto paradoxes such as Hardy’s paradox (Aharonov et
al.,2002; Lundeen and Steinberg, 2009; Yokota et al., 2009)and the
three-box paradox (Resch et al., 2004), as wellas the violation of
generalized Leggett-Garg inequalitiesthat indicate nonclassical
behavior (Dressel et al., 2011;Emary et al., 2014; Goggin et al.,
2011; Groen et al.,2013; Palacios-Laloy et al., 2010; Suzuki et
al., 2012).Conditioned averages have also been used to
experimen-tally measure physically meaningful quantities
includingsuperluminal group velocities in optical fiber (Brunneret
al., 2004), momentum-disturbance relationships in atwo-slit
interferometer (Mir et al., 2007), and locally av-eraged momentum
streamlines passing through a two-slitinterferometer (Kocsis et
al., 2011) [i.e., along the energy-momentum tensor field (Hiley and
Callaghan, 2012), orPoynting vector field (Bliokh et al., 2013;
Dressel et al.,2014)].
This Colloquium is structured as follows. In the nexttwo
sections we explain what a weak value is and how itappears in the
theory quite generally. We then explainhow it is possible to
measure both its real and imaginaryparts and explore the three
classes of experiments out-lined above that make use of weak
values. This approach
allows us to address the importance and utility of weakvalues in
a clear and direct way without stumbling overinterpretations that
have historically tended to obscurethese points. Throughout this
Colloqium, we make useof one simple notation for expressing
theoretical notions,and one experimental setup — a polarized beam
passingthrough a birefringent crystal.
II. WHAT IS A WEAK VALUE?
First introduced by Aharonov et al. (1988), weak val-ues are
complex numbers that one can assign to the pow-ers of a quantum
observable operator  using two states:an initial state |i〉,
called the preparation or preselection,and a final state |f〉,
called the postselection. The nthorder weak value of  has the
form
Anw =〈f |Ân|i〉〈f |i〉 , (1)
where the order n corresponds to the power of  thatappears in
the expression. In this Colloquium, we clar-ify how these peculiar
complex expressions appear nat-urally in laboratory measurements.
To accomplish thisgoal, we derive them in terms of measurable
detectionprobabilities. Weak values of every order appear whenwe
characterize how an intermediate interaction affectsthese detection
probabilities.
Consider a standard prepare-and-measure experiment.If a quantum
system is prepared in an initial state |i〉,the probability of
detecting an event corresponding tothe final state |f〉 is given by
the squared modulus oftheir overlap P = |〈f |i〉|2. If, however, the
initial stateis modified by an intermediate unitary interaction
Û(�),the detection probability also changes to P� = |〈f |i′〉|2
=|〈f |Û(�)|i〉|2.
In order to calculate the relative change between theoriginal
and the modified probability, we must examinethe unitary operator
Û(�) carefully. In quantum mechan-ics, any observable quantity is
represented by a Hermitianoperator. Stone’s theorem states that any
such Hermi-tian operator  can generate a continuous
transformationalong a complementary parameter � via the unitary
oper-ator Û(�) = exp(−i�Â). For instance, if  is an
angularmomentum operator, the unitary transformation gener-ates
rotations through an angle �, or if  is a Hamilto-nian, the
unitary operator generates translations alonga time interval �, and
so on. In (Aharonov et al., 1988)(and most subsequent appearances
of the weak value) Âis chosen to be an impulsive interaction
Hamiltonian ofproduct form; we return to this special case in
Section III.
If � is small enough, or in other words if Û(�) is “weak,”we
can consider its Taylor series expansion. The detec-tion
probability introduced above can then be written as
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3
(shown here to first order)
P� = |〈f |Û(�)|i〉|2 = |〈f |(1− i�Â+ . . . )|i〉|2
= P + 2� Im〈i|f〉〈f |Â|i〉+O(�2). (2)
As long as |i〉 and |f〉 are not orthogonal (i.e. P 6= 0),we can
divide both sides of Eq. (2) by P to obtain therelative correction
(shown here to second order):
P�P
= 1 + 2� ImAw − �2[ReA2w − |Aw|2
]+O(�3), (3)
where Aw is the first order weak value and A2w is the
second order weak value as defined in Eq. (1). Here wearrive at
our operational definition: weak values char-acterize the relative
correction to a detection proba-bility |〈f |i〉|2 due to a small
intermediate perturbationÛ(�) that results in a modified detection
probability|〈f |Û(�)|i〉|2. Although we show the expansion only
tosecond order here, we emphasize that the full Taylor se-ries
expansion for P�/P is completely characterized bycomplex weak
values Anw of all orders n (Di Lorenzo,2012; Dressel and Jordan,
2012d; Kofman et al., 2012).
When the higher order terms in the expansion (3) canbe
neglected, one has a linear relationship between theprobability
correction and the first order weak value,which we call the weak
interaction regime. These termscan be neglected under two
conditions: (a) the relativecorrection P�/P − 1 is itself
sufficiently small, and (b)�ImAw is sufficiently large compared to
the sum of higherorder corrections (Duck et al., 1989). When these
con-ditions do not hold (such as when P → 0), the termsinvolving
higher order weak values Anw become signifi-cant and can no longer
be neglected (Di Lorenzo, 2012).Most experimental work involving
weak values has beendone in the weak interaction regime
characterized by thefirst order weak value, so we will limit our
discussion tothat regime as well. In Section III, we put these
ideas inthe context of a real optics experiment and discuss howone
measures weak values in the laboratory.
III. HOW DOES ONE MEASURE A WEAK VALUE?
In general, weak values are complex quantities. In or-der to
determine a weak value, one must be able to mea-sure both its real
and imaginary parts. Here, we use anoptical experimental example to
show how one can mea-sure a complex weak value associated with a
polarizationobservable. Although this particular example can alsobe
understood using classical wave mechanics (Brunneret al., 2003;
Howell et al., 2010), the quantum mechanicalanalysis we provide
here has wider applicability.
Consider the setup shown in Fig. 1(a). A collimatedlaser beam is
prepared in an initial state |i〉|ψi〉, where |i〉is an initial
polarization state and |ψi〉 is the state of thetransverse beam
profile. The polarization is prepared
(a)(a)
(b)
(c)
Collimating Lens
Collimating Lens
HWP QWP
Polariser CCD
BirefringentCrystal
Position Imaging Lens
Momentum Imaging Lens
FIG. 1 An experiment for illustrating how one can measureweak
values. (a) A Gaussian beam from a single mode fiberis collimated
by a lens and prepared in an initial polarizationstate by a
quarter-wave plate (QWP) and half-wave plate(HWP). A polarizer
postselects the beam on a final polar-ization state. A CCD then
measures the position-dependentbeam intensity. (b) A birefringent
crystal is inserted betweenthe wave plates and polarizer to
displace different polariza-tions by a small amount. A lens images
the transverse posi-tion on the output face of the crystal onto the
CCD in orderto measure the real part of the polarization weak value
asa linear shift in the postselected intensity. (c) The lens
ischanged to imaging the far-field of the crystal face onto theCCD
as the transverse momentum in order to determine theimaginary part
of the polarization weak value (details in thetext).
through the use of a quarter-wave plate (QWP) and ahalf-wave
plate (HWP). The beam then passes througha linear polarizer aligned
to a final polarization state |f〉before impacting a charge coupled
device (CCD) imagesensor for a camera. Each pixel of the CCD
measures aphoton of this beam with a detection probability
givenby
P = |〈f |i〉|2|〈ψf |ψi〉|2, (4)
where |ψf 〉 is the final transverse state postselected byeach
pixel. For our purposes, this state corresponds toeither a specific
transverse position |ψf 〉 = |x〉 or trans-verse momentum |ψf 〉 =
|p〉, depending on whether weimage the position or the momentum
space onto the CCD[e.g., using a Fourier lens as shown in Fig.
1(c)]. We willrefer to this detection probability P as the
“unperturbed”probability.
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4
|i〉
|f〉0.2
0.1
φ
θ
FIG. 2 A band around the equator of the Poincaré sphereshowing
the initial polarization |i〉 (dot, back of sphere)from Eq. (8) and
postselection polarization |f〉 (dot, frontof sphere) from Eq. (9).
We also indicate the small anglesthat make |f〉 almost orthogonal to
|i〉.
We now introduce a birefringent crystal between thepreparation
wave plates and the postselection polarizer,as shown in Fig. 1(b).
The crystal separates the beaminto two beams with horizontal and
vertical polarizations.The transverse displacements depend on the
birefrin-gence properties of the crystal and on the crystal
length.We assume that the crystal is tilted with respect to
theincident beam so that each polarization component is dis-placed
by an equal amount � = τv where τ is the timespent inside the
crystal and v is the displacement speed.
The effect of the birefringent crystal can be expressed
by a time evolution operator Û(τ) = e−i τĤ/~ with aneffective
interaction Hamiltonian
Ĥ = vŜ ⊗ p̂. (5)
Here, Ŝ = |H〉〈H| − |V 〉〈V | is the Stokes polarizationoperator
that assigns eigenvalues +1 and −1 to the |H〉and |V 〉
polarizations, respectively, and p̂ is the trans-verse momentum
operator that generates translations inthe transverse position x.
This time evolution operatorÛ(τ) correlates the polarization
components of the beamwith their transverse position by translating
them in op-posite directions. Each pixel of the CCD then collects
aphoton with a “perturbed” probability given by
P� = |〈f |〈ψf |e−i�Ŝ⊗p̂/~|i〉|ψi〉|2, (6)
which has the form of Eq. (2) with the generic operatorÂ
replaced by the product operator Ŝ ⊗ p̂.
As a visual example, consider a Gaussian beam
〈x|ψi〉 = (2πσ2)−1/4 exp(− x
2
4σ2
), (7)
with an initial antidiagonal polarization preparation witha
slight ellipticity:
|i〉 = |H〉 − eiφ|V 〉√
2, φ = 0.1, (8)
that passes through a linear postselection polarizer thatis
oriented at a small angle (0.2 rad in this example) fromthe
diagonal state:
|f〉 = cos θ2|H〉+ sin θ
2|V 〉, θ = π
2− 0.2. (9)
These two nearly orthogonal polarization states areshown on a
band around the equator of the Poincarésphere in Fig. 2. Without
the crystal present [Fig. 1(a)],the CCD measures the initial
Gaussian intensity profileshown as a dashed line in Fig. 3(a) with
a total postse-lection probability given by |〈f |i〉|2 = 0.012. When
thecrystal is present [Fig. 1(b)], the orthogonal
polarizationcomponents become spatially separated by a
displace-ment � before passing through the postselection
polar-izer. The measured profiles for different crystal lengthsare
shown as the solid line distributions in Fig. 3(a). Thedotted line
distributions show the unperturbed (but stillpostselected) profiles
for comparison.
In the weak interaction regime, the crystal is short, �is small,
and the two orthogonally polarized beams aredisplaced by a small
amount before they interfere at thepostselection polarizer. As
shown in Section II, we canexpand the ratio between the perturbed
and unperturbedprobabilities to first order in � and isolate the
linear prob-ability correction term:
P�P− 1 ≈ 2τ
~ImHw (10)
=2�
~[ReSwImpw + ImSwRepw] .
Since the Hamiltonian from Eq. (5) is of product form,its first
order weak value contribution ImHw expands toa symmetric
combination of the real and imaginary partsof the weak values of
polarization Sw = 〈f |Ŝ|i〉/〈f |i〉 andmomentum pw = 〈ψf |p̂|ψi〉/〈ψf
|ψi〉. A clever choice ofpreselection and postselection states
therefore allows anexperimenter to isolate each of these quantities
using dif-ferent experimental setups (Aharonov et al., 1988;
Jozsa,2007; Shpitalnik et al., 2008).
To illustrate this idea for the polarization weak value,the
procedure for measuring the real part ReSw is shownin Fig. 1(b). We
image the output face of the crystal ontothe CCD so that each pixel
corresponds to a postselectionof the transverse position |ψf 〉 =
|x〉. As a result, themomentum weak value for each pixel becomes
pw =〈x|p̂|ψi〉〈x|ψi〉
=−i~∂xψi(x)
ψi(x)= i~
x
2σ2, (11)
using the Gaussian profile in Eq. (7).Since this expression is
purely imaginary, Eq. (10) sim-
plifies to
P�P≈ 1 + � x
σ2ReSw, (12)
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5
effectively isolating the quantity ReSw to first order in�. The
solid curves in Fig. 3(b) illustrate the ratio P�/Pas a function of
x for different values of �. When � issufficiently small, the
expansion of P�/P to first orderin Eq. (12) [dashed lines in Fig.
3(b)] is a good approx-imation over most of the beam profile.
Pragmatically,this means that one can average the whole beam
profileand still retain a linear correction that is proportional
toReSw, as done originally by Aharonov et al. (1988).
The analogous procedure for measuring the imaginarypart ImSw is
shown in Fig. 1(c). We image the Fourierplane of the crystal onto
the CCD so that each pixel cor-responds to a postselection of the
transverse momentum|ψf 〉 = |p〉. As a result, the momentum weak
value foreach pixel becomes simply
pw =〈p|p̂|ψi〉〈p|ψi〉
=p〈p|ψi〉〈p|ψi〉
= p. (13)
Since this expression is now purely real, Eq. (10) simpli-fies
to
P�P≈ 1 + �2p
~ImSw, (14)
effectively isolating the quantity ImSw to first order in�. As
with Eq. (12), this first order expansion is a goodapproximation
over most of the Fourier profile when �is sufficiently small.
Hence, the profile may be similarlyaveraged and retain the linear
correction proportional toImSw, as done originally in Aharonov et
al. (1988).
Note that we could also isolate the real and imagi-nary parts of
pw in a similar manner through a judiciouschoice of polarization
postselection states. More gener-ally, one can use this technique
to isolate weak values ofany desired observable by constructing
Hamiltonians ina product form such as Eq. (5) and cleverly choosing
thepreselection and postselection of the auxiliary degree
offreedom.
IV. HOW CAN WEAK VALUES BE USEFUL?
In Section III, we showed how the relative change
inpostselection probability can be completely described bycomplex
weak value parameters. We also elucidated howthe real and imaginary
parts of the first order weak valuecan be isolated and therefore
measured in the weak in-teraction regime.
In this section we focus on three main applications ofthe first
order weak value. First, we show how cleverchoices of the initial
and final postselected states can re-sult in large weak values that
can be used to sensitivelydetermine unknown parameters affecting
the state evolu-tion. Second, we show how the complex character of
theweak value may be used to directly determine a quan-tum state.
Third, we show how the real part of the weakvalue can be
interpreted as a form of conditioned averagepertaining to an
observable.
!10 !5 5 10x
1.5
3PΕ !10
!3"
Ε # 0.02
!10 !5 5 10x
0.8
1
1.2
1.4
PΕ#P
Ε # 0.02
!10 !5 5 10x
1.5
3
PΕ !10!3"
Ε # 0.1
!10 !5 5 10x
1
2
3
4PΕ#P
Ε # 0.1
!10 !5 5 10x
3.5
7PΕ !10
!3"
Ε # 0.5
!10 !5 5 10x
10
20
30
40
PΕ#P
Ε # 0.5
(a) (b)
FIG. 3 (a) Comparisons between perturbed profiles (solid,for
various values of beam displacement �) and a fixed un-perturbed
profile (dashed, corresponding to P ). Note thatboth curves
represent postselected measurements. (b) Theexact ratio of the two
curves (solid) is compared to the firstorder approximation
(dashed). When � is sufficiently small,the first order
approximation adequately models the quantityP�/P over most of the
profile.
A. Weak value amplification
In precision metrology an experimenter is interestedin
estimating a small interaction parameter, such as thetransverse
beam displacement � = τv due to the crys-tal in Section III. As the
first order approximation ofP/P� holds in the weak interaction
regime, the value of� can be directly determined. We briefly note
that theappearance of the joint weak value of Eq. (10) in a
pa-rameter estimation experiment is no accident: as pointedout by
Hofmann (2011), this quantity is the score usedto calculate the
Fisher information that determines theCramer-Rao bound for the
estimation of an unknown pa-rameter such as � (Helstrom, 1976;
Hofmann et al., 2012;Jordan et al., 2014; Knee and Gauger, 2014;
Pang et al.,2014; Viza et al., 2013).
Being able to resolve a small � in the presence of back-ground
noise requires the joint weak value factor in Eq.(10) to be
sufficiently large. When this weak value factoris large it will
amplify the linear response. Critically, theinitial and final
states for the weak values Sw and pw canbe strategically chosen to
produce a large amplificationfactor. This is the essence of the
technique used in weakvalue amplification (Dixon et al., 2009; Egan
and Stone,2012; Gorodetski et al., 2012; Hayat et al., 2013;
Hogan
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6
et al., 2011; Hosten and Kwiat, 2008; Jayaswal et al.,2014;
Kedem, 2012; Magana-Loaiza et al., 2013; Pfeiferand Fischer, 2011;
Puentes et al., 2012; Shomroni et al.,2013; Starling et al.,
2010a,b; Strübi and Bruder, 2013;Turner et al., 2011; Viza et al.,
2013; Xu et al., 2013;Zhou et al., 2013, 2012; Zilberberg et al.,
2011).
For a tangible example of how this amplification worksfor
estimating �, consider the measurement in Fig. 1(b).Averaging the
position recorded at every pixel producesthe centroid∫
xP�(x|θ) dx ≈〈x〉+ �(
〈x2〉/σ2)ReSw
1 + �(〈x〉 /σ2)ReSw, (15)
= �ReSw.
To compute Eq. (15) we used the perturbed conditionalprobability
P�(x|θ) = P�(x, θ)/
∫P�(x, θ)dx computed
from Eq. (12) as a function of the pixel position x, anda given
postselection polarization angle θ, as well as theGaussian moments
〈x〉 = 0 and
〈x2〉
= σ2 of the unper-turbed beam profile. Dividing the measured
centroid bythe (known) quantity ReSw allows us to determine
thesmall parameter �.
Alternatively, if the CCD measures the Fourier planeas in Fig.
1(c), then each pixel corresponds to a trans-verse momentum.
Finding the centroid in this case pro-duces ∫
pP�(p|θ) dp ≈〈p〉+ 2�
〈p2〉
ImSw/~1 + 2� 〈p〉 ImSw/~
(16)
= �~
2σ2ImSw,
where we used Eq. (14) and the Gaussian moments 〈p〉 =0 and
〈p2〉
= (~/2σ)2 of the unperturbed beam profile.The amplification
occurs in each case because the fac-
tor ReSw in Eq. (15) or 2〈p2〉
ImSw in Eq. (16) can bemade large by clever choices of
polarization postselection.For our example states [Eqs. (8) and
(9)], the polarizationweak value is Sw = 〈f |Ŝ|i〉/〈f |i〉 ≈ 7.5 +
3.2i. Notably,both the real and imaginary parts of the weak value
inthis case are larger than 1, which is the maximum eigen-value of
Ŝ. The plot in Fig. 4(a) shows how the real andimaginary parts of
the weak value vary with the choiceof postselection angle θ.
One cannot obtain such amplification to the sensitivityfor free,
however. As the weak value factor Sw becomeslarge, the detection
probability necessarily decreases, asshown in Fig. 4(b). Hence, the
weak interaction approx-imation that assumes 2�Im(S ⊗ p)w � |〈f
|i〉|2|〈ψf |ψi〉|2for each pixel will eventually break down and it
will benecessary to include higher-order terms in � that havebeen
neglected, spoiling the linear response (Cho et al.,2010; Di
Lorenzo, 2012; Dressel and Jordan, 2012b,d;Geszti, 2010; Kofman et
al., 2012; Koike and Tanaka,2011; Nakamura et al., 2012; Pan and
Matzkin, 2012;Parks and Gray, 2011; Shikano and Hosoya, 2010,
2011;
p 2 pq
-10
10
20Sw
p2 p 2 p
q
0.5
1PHqL
p2
2.5
12H10-3L
(b)(a)
FIG. 4 (a) Real (dashed) and imaginary (solid) parts of the
polarization weak value Sw = 〈f |Ŝ|i〉/〈f |i〉, with initial
state|i〉 given in Eq. (8) and shown in Fig. 2, and final state
|f〉that depends on a varying angle θ. The eigenvalue bounds of±1
are shown as dotted lines for reference, while the dots in-dicate
the final state chosen in Eq. (9). (b) The postselectionprobability
P (θ) = |〈f |i〉|2 as a function of θ, showing howa large weak value
corresponds to a small detection proba-bility. The inset shows the
small probability region enlargedfor clarity, while the dots
similarly indicate the final state inEq. (9).
Susa et al., 2012; Wu and Li, 2011; Wu and Żukowski,2012; Zhu
et al., 2011). Moreover, the resulting low de-tection rate (i.e.,
collected beam intensity) make it dif-ficult to detect the signal,
leading to longer collectiontimes in order to overcome the noise
floor. Indeed, acareful analysis shows that the signal-to-noise
ratio fordetermining � within a fixed time duration remains
con-stant as the amplification increases (Feizpour et al.,
2011;Ferrie and Combes, 2014; Jordan et al., 2014; Knee andGauger,
2014; Starling et al., 2009)—the signal gainedby increasing the
amplification factors in Eq. (15) or (16)will exactly cancel the
uncorrelated shot noise gained bydecreasing the detection rate. The
scheme can also besensitive to decoherence during the measurement
(Kneeet al., 2013).
Nevertheless, there are two distinct advantages to us-ing this
amplification technique: (1) the detector collectsa fraction of the
total beam power due to the postselec-tion polarizer yet still
shows similar sensitivity to opti-mal estimation methods (Jordan et
al., 2014; Knee andGauger, 2014; Pang et al., 2014), and (2) the
weaknessof the measurement itself makes the amplification
robustagainst certain types of additional technical noise (suchas
1/f noise) (Feizpour et al., 2011; Ferrie and Combes,2014; Jordan
et al., 2014; Knee and Gauger, 2014; Star-ling et al., 2009). The
former advantage allows less ex-pensive equipment to be used, while
simultaneously en-abling the uncollected beam power to be
redirected else-where for other purposes (Dressel et al., 2013;
Starlinget al., 2010a). The latter advantage allows one to
amplifythe signal without also amplifying certain types of
unre-lated (but common) technical noise backgrounds. Thesetwo
advantages combined are precisely what has permit-ted experiments
such as (Dixon et al., 2009; Egan andStone, 2012; Hogan et al.,
2011; Hosten and Kwiat, 2008;
-
7
Jayaswal et al., 2014; Magana-Loaiza et al., 2013; Pfeiferand
Fischer, 2011; Starling et al., 2010a,b; Turner et al.,2011; Xu et
al., 2013; Zhou et al., 2013, 2012) to achievesuch phenomenal
precision with relatively modest labo-ratory equipment.
B. Measurable complex value
Since weak values are measurable complex quantities,they can be
used to directly measure other normally in-accessible complex
quantities in the quantum theory thatcan be expanded into sums and
products of complex weakvalues, such as the geometric phase
(Kobayashi et al.,2010, 2011; Sjöqvist, 2006). Most notably, one
can “di-rectly” measure the quantum state itself using this
tech-nique (Fischbach and Freyberger, 2012; Kobayashi et al.,2013;
Lundeen and Bamber, 2012; Lundeen et al., 2011;Malik et al., 2014;
Massar and Popescu, 2011; Salvailet al., 2013; Wu, 2013; Zilberberg
et al., 2011). Conven-tionally, a quantum state is determined
through the in-direct process of quantum tomography (Altepeter et
al.,2005). Like its classical counterpart, quantum tomogra-phy
involves making a series of projective measurementsin different
bases of a quantum state. This process is in-direct in that it
involves a time consuming postprocess-ing step where the density
matrix of the state must beglobally reconstructed through a
numerical search overthe alternatives consistent with the measured
projec-tive slices. Propagating experimental error through
thisreconstruction step can be problematic, and the com-putation
time can be prohibitive for determining high-dimensional quantum
states, such as those of orbital an-gular momentum.
We can bypass the need for such a global reconstruc-tion step by
expanding individual components of a quan-tum state directly in
terms of measurable weak values.For a simple example, we determine
the complex compo-nents of the initial polarization state |i〉 from
Section III,as expanded in the weak measurement basis {|H〉, |V
〉}.This is accomplished by the insertion of the identity
andmultiplication by a strategically chosen constant factorc =
〈D|H〉/〈D|i〉 = 〈D|V 〉/〈D|i〉, where the postselec-tion state |D〉 is
unbiased with respect to both |H〉 and|V 〉. With this clever choice
the scaled state has the form
c|i〉 = 〈D|H〉〈H|i〉〈D|i〉︸ ︷︷ ︸Hw
|H〉+ 〈D|V 〉〈V |i〉〈D|i〉︸ ︷︷ ︸Vw
|V 〉. (17)
That is, each complex component of the scaled state c|i〉can be
directly measured as a complex first order weakvalue. After
determining these complex components ex-perimentally, the state can
be subsequently renormalizedto eliminate the constant c up to a
global phase.
Furthermore, we can write the projections as |H〉〈H| =(1̂ + Ŝ)/2
and |V 〉〈V | = (1̂ − Ŝ)/2, so we can rewrite
the required weak values Hw = (1 + Sw)/2 and Vw =(1−Sw)/2 in
terms of the single polarization weak valueSw. We showed earlier
how to isolate and measure boththe real and imaginary parts of this
polarization weakvalue. Thus, we can completely determine the state
|i〉after the polarization weak value Sw has been measuredusing the
special postselection |D〉.
The primary benefit of this direct state estimation ap-proach is
that minimal postprocessing (and thus mini-mal experimental error
propagation) is required to re-construct individual state
components from the experi-mental data. The real and imaginary
parts of each purestate component in a desired basis directly
appear in thelinear response of a measurement device up to
appropri-ate scaling factors. Mixed states can also be measuredin a
similar way by scanning the postselection across amutually unbiased
basis, which will determine the Diracdistribution for the state
instead (Lundeen and Bamber,2012, 2014; Salvail et al., 2013); this
distribution is re-lated to the density matrix via a Fourier
transform.
The downside of this approach is that the denomina-tor 〈D|i〉 in
the constant c cannot become too small orthe linear approximation
used to measure Sw will breakdown (Haapasalo et al., 2011), causing
estimation errors(Maccone and Rusconi, 2014). This restriction
limitsthe generality of the technique for faithfully estimating
atruly unknown state. Furthermore, improperly calibrat-ing the weak
interaction can introduce unitary errors orproduce additional
decoherence that does not appear inprojective tomography
techniques. Nevertheless, the di-rect measurement technique can be
useful for determin-ing the components of most states.
C. Conditioned average
As our final example of the utility of weak values, weshow that
the real part of a weak value can be inter-preted as a form of
conditioned average associated withan observable. To show this we
first consider how eachpixel records polarization information in
the absence ofpostselection. After summing over all
complementarypostselections |f〉 in the perturbed probability P�(x,
f)in Eq. (6), we can express the total perturbed pixel prob-ability
as
P�(x) =∑f
|〈f |〈x|e−i�Ŝ⊗p̂/~|i〉|ψi〉|2 = 〈i|P̂x|i〉, (18)
in terms of a probability operator
P̂x = |〈x− �|ψi〉|2 |H〉〈H|+ |〈x+ �|ψi〉|2 |V 〉〈V |,= |〈x−
�Ŝ|ψi〉|2. (19)
The second line is a formal way of writing the proba-bility
operator more compactly in terms of the spectralrepresentation of
Ŝ. This formal expression also supports
-
8
Π 2 ΠΘ
#10
#5
5
10Sw
Ε % 6
Π 2 ΠΘ
#10
#5
5
10Sw
Ε % 1
Π 2 ΠΘ
#10
#5
5
10Sw
Ε % 0.5
Π 2 ΠΘ
#10
#5
5
10Sw
Ε % 0.1
(a) (b)
(c) (d)
FIG. 5 Conditioned average (22) of generalized
polarizationeigenvalues x/� for various values of the crystal
length �, usingthe beam profile illustrated in Figure 2. For large
� the aver-age is a classical conditioned average constrained to
the eigen-value range (dotted lines). For sufficiently small �,
however,the conditioned average (solid lines) approximates the
realpart (dashed lines) of the polarization weak value in Fig.
4.
the intuition that P̂x indicates that the crystal interac-tion
shifts the initial profile |〈x|ψi〉|2 of the beam by anamount that
depends on the polarization.
An experimenter can then assign a value of (x/�) toeach pixel x
and average those values over the perturbedprofile in Eq. (18) to
obtain the average polarization∫
x
�P�(x) dx = 〈i|Ŝ|i〉 (20)
for any preparation state |i〉. The values (x/�) assignedto each
pixel act as generalized eigenvalues for the po-larization operator
Ŝ (Dressel et al., 2010; Dressel andJordan, 2012a,c). An
experimenter must assign thesevalues in place of the standard
polarization eigenvaluesof ±1 because the pixels are only weakly
correlated withthe polarization. Although the values (x/�)
generally liewell outside the eigenvalue range of Ŝ, their
experimen-tal average in Eq. (20) always produces a sensible
averagepolarization.
The state independence of this procedure can be em-phasized by
noting that the assignment of the generalizedeigenvalues (x/�)
formally produces an operator identity,∫
x
�P̂x dx = Ŝ (21)
in terms of the probability operators P̂x in (19) that
cor-respond to each measured pixel. This identity guaranteesthat
the experimenter can faithfully reconstruct informa-tion about the
observable Ŝ for any unknown state byproperly weighting the
probabilities for measuring eachCCD pixel. In the special case of a
projective measure-ment, the probability operators will be the
spectral pro-jections for Ŝ and the assigned values will be the
eigenval-ues of Ŝ, which makes Eq. (21) a natural
generalization
of the spectral expansion of Ŝ to a generalized
measuringapparatus.
It is worth noting that since there are more pixels
thanpolarization eigenvalues, one can form an operator iden-tity
such as Eq. (21) in many different ways by assigningdifferent
values α(x) to the pixel probabilities. In such acase, the
information redundancy in the pixel probabili-ties gives the
freedom to choose appropriate values thatstatistically converge
more rapidly to the desired mean(Dressel et al., 2010; Dressel and
Jordan, 2012a,c). Forour purposes here, however, we use the
simplest genericchoice α(x) = x/�.
Including the effect of the postselection polarizer |f〉changes
this general result. The added polarizer con-ditions the total
pixel probability of Eq. (18). After as-signing the same
generalized polarization eigenvalues x/�to each pixel and averaging
these values over the condi-tioned profile, an experimenter will
find the conditionedaverage ∫
x
�P�(x|f) dx = Re
〈f |Ŝ|i〉〈f |i〉 +O(�
2). (22)
As shown in Eq. (15) this conditioned average of general-ized
polarization eigenvalues approximates the real partof a weak value
for small � in an experimentally mean-ingful way.
Importantly, even when � is not small the full con-ditioned
average of generalized eigenvalues (22) willsmoothly interpolate
between the weak value approxi-mation and a classical conditioned
average of polariza-tion. In Fig. 5 we illustrate this
interpolation for differ-ent values of �. This smooth
correspondence is essentialfor associating the experimental average
Eq. (22) to thepolarization Ŝ in any meaningful way. Indeed, we
haveshown (Dressel and Jordan, 2012b,d) that this interpola-tion
exactly describes how the initial polarization statedecoheres into
a classical polarization state with increas-ing measurement
strength. Moreover, this technique ofconstructing conditioned
averages of generalized eigen-values works quite generally for
other detectors (Dresselet al., 2011, 2012; Goggin et al., 2011;
Kedem and Vaid-man, 2010; Pryde et al., 2005; Romito et al., 2008;
Silvaet al., 2014; Weston et al., 2013; Zilberberg et al., 2013)and
produces similar interpolations between a classicalconditioned
average and the real part of a weak value.
The link between weak values and conditioned aver-ages has been
used to address several quantum para-doxes, such as Hardy’s paradox
(Aharonov et al., 2002;Lundeen and Steinberg, 2009; Yokota et al.,
2009) andthe three-box paradox (Resch et al., 2004). Anoma-lously
large weak values provide a measurable windowinto the inner
workings of these paradoxes by indicat-ing when quantum observables
cannot be understood inany classical way as properties related to
their eigenval-ues. Similarly, anomalously large weak values have
been
-
9
linked to violations of generalized Leggett-Garg inequal-ities
(Dressel et al., 2011; Emary et al., 2014; Gogginet al., 2011;
Groen et al., 2013; Palacios-Laloy et al.,2010; Suzuki et al.,
2012; Williams and Jordan, 2008)that indicate nonclassical (or
invasive) behavior in mea-surement sequences. This link has also
been exploited toprovide an experimental method for determining
phys-ically meaningful conditioned quantities, such as
groupvelocities in optical fibers (Brunner et al., 2004), or
themomentum-disturbance relationships for a two-slit
inter-ferometer (Mir et al., 2007).
A particularly notable experimental demonstration ofthe
connection between weak values and physically mean-ingful
conditioned averages is the measurement of thelocally averaged
momentum streamlines pB(x) passingthrough a two-slit interferometer
performed by Kocsiset al. (2011) using the weak value identity
Re〈x|p̂|ψi〉〈x|ψi〉
= ∂xΦ(x) = pB(x), (23)
where 〈x|ψi〉 = |〈x|ψi〉| exp[iΦ(x)/~] is the polar decom-position
of the initial transverse profile. This phase gra-dient has
appeared historically in Madelung’s hydrody-namic approach to
quantum mechanics (Madelung, 1926,1927), Bohm’s causal model (Bohm,
1952a,b; Traversaet al., 2013; Wiseman, 2007), the momentum part
ofthe local energy-momentum tensor (Hiley and Callaghan,2012), and
even the Poynting vector field of classical elec-trodynamics
(Bliokh et al., 2013; Dressel et al., 2014).Importantly, the weak
value connection provides thisquantity with an experimentally
meaningful definition asa weakly measured conditioned average.
V. CONCLUSIONS
In this Colloquium we explored how the quantum weakvalue
naturally appears in laboratory situations. We op-erationally
defined weak values as complex parametersthat completely
characterize the relative corrections todetection probabilities
that are caused by an intermedi-ate interaction. When the
interaction is sufficiently weak,these relative corrections can be
well approximated byfirst order weak values.
Using an optical example of a polarized beam passingthrough a
birefringent crystal, we showed how to use aproduct interaction to
isolate and measure both the realand imaginary parts of first order
weak values. This ex-ample allowed us to discuss three distinct
roles that thefirst order weak value has played in recent
experiments.
First, we showed how a large weak value can be usedto amplify a
signal used to sensitively estimate an un-known interaction
parameter in the (linear) weak inter-action regime. Although the
signal-to-noise ratio remainsconstant from this amplification due
to a correspondingreduction in detection probability, the technique
allows
one to amplify the signal above other technical noisebackgrounds
using fairly modest laboratory equipment.
Second, we showed that since the first order weak valueis a
measurable complex parameter, it can be used to ex-perimentally
determine other complex theoretical quan-tities. Notably, we showed
how the components of a purequantum state may be directly
determined up to a globalphase by measuring carefully chosen weak
values.
Third, we discussed the relationship between the realpart of a
first order weak value and a conditioned aver-age for an
observable. By conditionally averaging gener-alized eigenvalues for
the observable, we showed that oneobtains an average that smoothly
interpolates betweena classical conditioned average and a weak
value as theinteraction strength changes.
We have emphasized the generality of the quantumweak value as a
tool for describing experiments. Becauseof this generality, we
anticipate that many more applica-tions of the weak value will be
found in time. We hopethis Colloquium will encourage further
exploration alongthese lines.
ACKNOWLEDGMENTS
Acknowledgments.—JD and ANJ acknowledge supportfrom the National
Science Foundation under Grant No.DMR-0844899, and the US Army
Research Office un-der Grant No. W911NF-09-0-01417. MM, FMM, andRWB
acknowledge support from the US DARPA InPhoprogram. FMM and RWB
acknowledge support fromthe Canada Excellence Research Chairs
Program. MMacknowledges support from the European Commissionthrough
a Marie Curie fellowship. The authors thankJonathan Leach for
helpful discussions.
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Colloquium: Understanding Quantum Weak Values: Basics and
ApplicationsAbstract ContentsI IntroductionII What is a weak
value?III How does one measure a weak value?IV How can weak values
be useful?A Weak value amplificationB Measurable complex valueC
Conditioned average
V Conclusions Acknowledgments References