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;Strubi 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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2this technique include the measurement of 1A 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 63W of power postselected from 3.5mW to-tal beam power (Dixon et al., 2009), and a frequencysensitivity of 129kHz/

Hz obtained with 85W 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;Sjoqvist, 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 Hardys 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 A 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 A has the form

Anw =f |An|if |i , (1)

where the order n corresponds to the power of A 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 U(),the detection probability also changes to P = |f |i|2 =|f |U()|i|2.

In order to calculate the relative change between theoriginal and the modified probability, we must examinethe unitary operator U() carefully. In quantum mechan-ics, any observable quantity is represented by a Hermitianoperator. Stones theorem states that any such Hermi-tian operator A can generate a continuous transformationalong a complementary parameter via the unitary oper-ator U() = exp(iA). For instance, if A is an angularmomentum operator, the unitary transformation gener-ates rotations through an angle , or if A 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) Ais 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 U() is weak,we can consider its Taylor series expansion. The detec-tion probability introduced above can then be written as

3(shown here to first order)

P = |f |U()|i|2 = |f |(1 iA+ . . . )|i|2= P + 2 Imi|ff |A|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):

PP

= 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 perturbationU() that results in a modified detection probability|f |U()|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|i, where |iis an initial polarization state and |i 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 |fbefore 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|f |i|2, (4)

where |f is the final transverse state postselected byeach pixel. For our purposes, this state corresponds toeither a specific transverse position |f = |x or trans-verse momentum |f = |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 unperturbedprobability.

4|i

|f0.2

0.1

FIG. 2 A band around the equator of the Poincare 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 = v where 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 U() = ei H/~ with aneffective interaction Hamiltonian

H = vS p. (5)

Here, S = |HH| |V V | is the Stokes polarizationoperator that assigns eigenvalues +1 and 1 to the |Hand |V polarizations, respectively, and p is the trans-verse momentum operator that generates translations inthe transverse position x. This time evolution operatorU() 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 |f |eiSp/~|i|i|2, (6)which has the form of Eq. (2) with the generic operatorA replaced by the product operator S p.

As a visual example, consider a Gaussian beam

x|i = (2pi2)1/4 exp( x

2

42

), (7)

with an initial antidiagonal polarization preparation witha slight ellipticity:

|i = |H ei|V

2, = 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 2|H+ sin

2|V , = pi

2 0.2. (9)

These two nearly orthogonal polarization states areshown on a band around the equator of the Poincaresphere 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:

PP 1 2

~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 |S|i/f |i andmomentum pw = f |p|i/f |i. 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 |f = |x. As a result, themomentum weak value for each pixel becomes

pw =x|p|ix|i =

i~xi(x)i(x)

= i~x

22, (11)

using the Gaussian profile in Eq. (7).Since this expression is purely imaginary, Eq. (10) sim-

plifies to

PP 1 + x

2ReSw, (12)

5effectively 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|f = |p. As a result, the momentum weak value foreach pixel becomes simply

pw =p|p|ip|i =

pp|ip|i = p. (13)

Since this expression is now purely real, Eq. (10) simpli-fies to

PP 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

3P !10!3"

# 0.02

!10 !5 5 10x0.8

1

1.2

1.4P#P

# 0.02

!10 !5 5 10x

1.5

3P !10!3"

# 0.1

!10 !5 5 10x1

2

34

P#P

# 0.1

!10 !5 5 10x

3.5

7P !10!3"

# 0.5

!10 !5 5 10x10203040P#P

# 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 = v 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

6et 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; Strubi 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|) dx x+ (x2 /2)ReSw

1 + (x /2)ReSw , (15)

= ReSw.

To compute Eq. (15) we used the perturbed conditionalprobability P(x|) = P(x, )/

P(x, )dx computed

from Eq. (12) as a function of the pixel position x, anda given postselection polarization angle , as well as theGaussian moments x = 0 and x2 = 2 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|) dp p+ 2 p2 ImSw/~

1 + 2 p ImSw/~ (16)

= ~

22ImSw,

where we used Eq. (14) and the Gaussian moments p =0 and

p2

= (~/2)2 of the unperturbed beam profile.The amplification occurs in each case because the fac-

tor ReSw in Eq. (15) or 2p2

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 |S|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 S. The plot in Fig. 4(a) shows how the real andimaginary parts of the weak value vary with the choiceof postselection angle .

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 2Im(S p)w |f |i|2|f |i|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

1PHqLp2

2.5

12H10-3L

(b)(a)

FIG. 4 (a) Real (dashed) and imaginary (solid) parts of the

polarization weak value Sw = f |S|i/f |i, with initial state|i given in Eq. (8) and shown in Fig. 2, and final state |fthat depends on a varying angle . The eigenvalue bounds of1 are shown as dotted lines for reference, while the dots in-dicate the final state chosen in Eq. (9). (b) The postselectionprobability P () = |f |i|2 as a function of , 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 Zukowski,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;

7Jayaswal 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; Sjoqvist, 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|HH|iD|i Hw

|H+ D|V V |iD|i Vw

|V . (17)

That is, each complex component of the scaled state c|ican 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 |HH| =(1 + S)/2 and |V V | = (1 S)/2, so we can rewrite

the required weak values Hw = (1 + Sw)/2 and Vw =(1Sw)/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 |iafter 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|eiSp/~|i|i|2 = i|Px|i, (18)

in terms of a probability operator

Px = |x |i|2 |HH|+ |x+ |i|2 |V V |,= |x S|i|2. (19)

The second line is a formal way of writing the proba-bility operator more compactly in terms of the spectralrepresentation of S. This formal expression also supports

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(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 Px indicates that the crystal interac-tion shifts the initial profile |x|i|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|S|i (20)

for any preparation state |i. The values (x/) assignedto each pixel act as generalized eigenvalues for the po-larization operator S (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 S, 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

Px dx = S (21)

in terms of the probability operators Px in (19) that cor-respond to each measured pixel. This identity guaranteesthat the experimenter can faithfully reconstruct informa-tion about the observable S 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 S and the assigned values will be the eigenval-ues of S, which makes Eq. (21) a natural generalization

of the spectral expansion of S 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 (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 (x) = x/.

Including the effect of the postselection polarizer |fchanges 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 |S|if |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 S 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 Hardys 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

9linked 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

Rex|p|ix|i = x(x) = pB(x), (23)

where x|i = |x|i| exp[i(x)/~] is the polar decom-position of the initial transverse profile. This phase gra-dient has appeared historically in Madelungs hydrody-namic approach to quantum mechanics (Madelung, 1926,1927), Bohms 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