Complex counterpart of variance in quantum measurements for pre- and post-selected systems
Kazuhisa Ogawa, Natsuki Abe, Hirokazu Kobayashi, Akihisa Tomita
aa r X i v : . [ qu a n t - ph ] F e b Complex counterpart of variance in quantum measurementsfor pre- and postselected systems
Kazuhisa Ogawa, ∗ Natsuki Abe, Hirokazu Kobayashi, and Akihisa Tomita Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan School of System Engineering, Kochi University of Technology, Tosayamada-cho, Kochi 782-8502, Japan (Dated: February 15, 2021)The variance of an observable for a preselected quantum system, which is always real and non-negative, appears in the increase of the probe wave packet width in indirect measurement. We extendthis framework to pre- and postselected systems, and formulate a complex-valued counterpart ofthe variance called “weak variance.” In our formulation, the real and imaginary parts of the weakvariance appear in the changes in the probe wave packet width in vertical-horizontal and diagonal-antidiagonal directions on the quadrature phase plane, respectively. Using an optical system, weexperimentally demonstrate these changes in a probe wave packet width caused by the real negativeand pure imaginary weak variances. Furthermore, we describe that the weak variance can beexpressed as the variance of the weak-valued probability distribution for pre- and post-selectedsystems. These operational and statistical interpretations support that our formulation of the weakvariance is reasonable as a complex counterpart of the variance for pre- and post-selected systems.
I. INTRODUCTION
In quantum measurements, measurement outcomesshow probabilistic behavior. This characteristic is pe-culiar not seen in classical systems, and has been theroot of many fundamental arguments in quantum theory[1]. In the quantum measurement of observable ˆ A , theprobabilistic behavior of its outcomes is characterized bythe statistics such as expectation value h ˆ A i and variance σ ( ˆ A ). These values are generally measured accordingto the indirect measurement model [2]. In indirect mea-surement, the target system to be measured is coupledwith an external probe system by von Neumann inter-action. Regardless of the coupling strength, the expec-tation value h ˆ A i and the variance σ ( ˆ A ) for the targetsystem are obtained from the displacement of the probewave packet and the increase of its width, respectively. Inother words, the probe wave packet in the indirect mea-surement serves as the interface that displays the prob-abilistic characteristics of the target system in quantummeasurement.It is interesting that, when the target system is furtherpostselected, the displacement of the probe wave packetshows a different value from the expectation value h ˆ A i .Especially, when the coupling strength is weak (weakmeasurement setup), the probe displacement is given by h ˆ A i w := h f | ˆ A | i i / h f | i i for the pre- and postselected states {| i i , | f i} , which was formulated as the weak value [3].The weak value is complex in general and can exceed thespectral range of ˆ A . Nevertheless, by regarding the weakvalue as a complex counterpart of the expectation valuefor the pre- and postselected system, new approaches ∗ [email protected] to fundamental problems in quantum mechanics involv-ing pre- and postselection has been investigated, suchas various quantum paradoxes [4–10], understanding ofthe violation of Bell inequality using negative probabili-ties [11], the relationship between disturbance and com-plementarity in quantum measurements [12–14], verifica-tion of the uncertainty relations [15–17], observation ofBohmian trajectories [18, 19], and demonstration of theviolation of macrorealism [20, 21].Similar to the relation between the weak value and theexpectation value, does there exist a counterpart of thevariance for pre- and postselected systems? The key tothis research is the function of the probe wave packet inindirect measurement as an interface that displays thecharacteristics of the target system. As mentioned ear-lier, the variance σ ( ˆ A ) for a preselected system appearsin the increase of the probe wave packet width in indi-rect measurement, and due to the non-negativity of thevariance, the wave packet width never decreases. Forpre- and postselected systems, on the other hand, anycounterparts of the variance cannot be observed in thetypical framework of the weak measurement [3], in whichthe probe wave packet width does not change because thesecond- and higher-order terms of the coupling strengthare ignored. Here, we focus on the recent studies report-ing that when considering the second- and higher-orderterms of the coupling strength the probe wave packetwidth can not only increase but also decrease under ap-propriate pre- and postselection [22, 23]. If reinterpretingthe phenomena in these reports as the consequence ofa counterpart of the variance for pre- and postselectedsystems, it may be possible to formulate an effectivevariance-like quantity that can be negative.In this study, we investigate the general changes inprobe wave packet width in indirect measurements forpre- and postselected systems, and based on this prop-erty, formulate a counterpart of the variance for the pre-and postselected systems. This counterpart, which wecall weak variance , can indeed be negative and is ob-served as the decrease of the probe wave packet width.Moreover, the weak variance is complex in general, andcan be understood by the superordinate concept of thechange in the probe wave packet width on the quadra-ture phase plane. To demonstrate this, we performed anoptical experiment observing the changes in the beampacket width in proportion to the real and imaginaryparts of the complex weak variance. In addition, to helpunderstand the outline of the weak variance, we introducethe expression of the weak variance as the second-ordermoment of the weak-valued probability distribution [4–13, 24–26], which is a quasi-probability distribution forpre- and postselected systems. From this agreement be-tween the operational and statistical interpretation, ourformulation of the weak variance can be considered to bea reasonable definition as a complex counterpart of thevariance for pre- and postselected systems, compared toother formulations proposed previously [27–39]. Further-more, we also formulate a counterpart of the higher-ordermoment, and investigate their operational and statisticalmeanings and applications. II. WEAK VARIANCE APPEARING ININDIRECT MEASUREMENT FOR PRE- ANDPOSTSELECTED SYSTEMS
We first review the indirect measurement procedureusing a Gaussian probe shown in Figs. 1(a) and (d), andexplain that the complex weak variance appears in thelatter system. The target system to be measured andthe probe system are preselected in | i i and | φ i , respec-tively. The initial probe state | φ i can be expanded as | φ i = R ∞−∞ d Xφ ( X ) | X i , where the wavefunction φ ( X )is the Gaussian distribution φ ( X ) = π − / exp( − X / X is a dimensionless variable [40]. The observ-able of dimensionless position ˆ X can be spectrally de-composed as ˆ X = R ∞−∞ d XX | X ih X | . The time evolu-tion by the interaction Hamiltonian ˆ A ⊗ ˆ K is representedby the unitary operator ˆ U ( θ ) = exp( − i θ ˆ A ⊗ ˆ K ), whereˆ A = P j a j ˆ Π j is the observable to be measured of thetarget system, a j is an eigenvalue of ˆ A , ˆ Π j is the projec-tor onto the eigenspace of ˆ A belonging to the eigenvalue a j , ˆ K is the canonical conjugate observable of ˆ X thatsatisfies [ ˆ X, ˆ K ] = iˆ1, and θ is a parameter with the recip-rocal dimension of ˆ A . θ k ˆ A k ( k ˆ A k is the largest eigenvalueof ˆ A ) characterizes the coupling strength: if θ k ˆ A k ≫ ≪ systemProbe P r o b a b ili t y d e n s i t y Variance (a) (b)(c)(d) P r o b a b ili t y d e n s i t y P r o b a b ili t y d e n s i t y P r o b a b ili t y d e n s i t y Variance Variance Variance
Strong conditionWeak condition (e) P r o b a b ili t y d e n s i t y P r o b a b ili t y d e n s i t y Variance
Weak measurement
TargetsystemsystemProbeTargetsystem
Variance : Postselection
FIG. 1. (a) Quantum circuit of indirect measurement for thepreselected system | i i . (b) Change of the probe wave packetcaused by the interaction in the quantum circuit (a) underthe strong coupling condition ( θ k ˆ A k ≫ A for | i i . (c) Change of the probe wave packetunder the weak coupling condition ( θ k ˆ A k ≪ σ ( ˆ A ), but never decreases. (d) Quantumcircuit of indirect measurement for the pre- and postselectedsystem {| i i , | f i} (weak measurement setup). (e) Change of theprobe wave packet in weak measurement circuit (d). The realpart of the weak variance appears in the variance change ofthe probe wave packet after the postselection. Unlike the pre-selected system (c), the variance of the probe wave packet candecrease when the real part of the weak variance is negative. able ˆ A as shown in Fig. 1 (a). Suppose that we performˆ X measurement in the probe system to the state afterthe interaction | Ψ i := exp( − i θ ˆ A ⊗ ˆ K ) | i i| φ i . The prob-ability distribution P ( X ) of obtaining the result X isrepresented as P ( X ) = |h X | Ψ i| = X j p j | φ ( X − θa j ) | , (1)where p j := h i | ˆ Π j | i i is the projection probability of | i i onto ˆ Π j . If the coupling is strong ( θ k ˆ A k ≫ | φ ( X − θa j ) | for each j is well separated from eachother and P ( X ) reproduces the probability distribution { p j } j as shown in Fig. 1(b). On the other hand, if thecoupling is weak ( θ k ˆ A k ≪ P ( X ) does not reproduce { p j } j asshown in Fig. 1(c). Nevertheless, regardless of the cou-pling strength, the statistics of ˆ A for the target system | i i , such as the expectation value h ˆ A i and the variance σ ( ˆ A ), can be acquired from the changes of the probedistribution P ( X ). The expectation value and the vari-ance of X for P ( X ), h ˆ X i f and σ ( ˆ X ), are respectivelyexpressed as h ˆ X i f = h ˆ X i i + θ h ˆ A i , σ ( ˆ X ) = σ ( ˆ X ) + θ σ ( ˆ A ) . (2) h ˆ X i i and σ ( ˆ X ) are the expectation value and the vari-ance of ˆ X for the initial probe state | φ i , respectively,and in this case, h ˆ X i i = 0 and σ ( ˆ X ) = 1 /
2. There-fore, the expectation value h ˆ A i and the variance σ ( ˆ A )can be measured under both strong and weak couplingconditions. We here stress that the variance of the probewave packet after the interaction σ ( ˆ X ) never decreasesdue to the non-negativity of the variance σ ( ˆ A ).We next consider that the target system is pre- andpostselected in | i i and | f i , respectively, as in Fig. 1(d).The unnormalized state of the probe system after thepost-selection | ˜ φ f i := h f | Ψ i is represented as | ˜ φ f i = h f | i i (cid:18) ˆ1 − i θ h ˆ A i w ˆ K − θ h ˆ A i w ˆ K (cid:19) | φ i + O ( θ ) . (3)The expectation value of ˆ X for this unnormalized state | ˜ φ f i is h ˆ X i f = h ˜ φ f | ˆ X | ˜ φ f i / h ˜ φ f | ˜ φ f i = Re h ˆ A i w θ + O ( θ ),where the real part of the weak value h ˆ A i w = h f | ˆ A | i i / h f | i i appears in the displacement of the probe wave packet, asknown in the weak measurement [3]. The imaginary partof the weak value is observed in the displacement of theprobe wave packet in ˆ K basis: h ˆ K i f = Im h ˆ A i w θ + O ( θ )[41]. By introducing the generalized position operatorˆ M := ˆ X cos α + ˆ K sin α ( α ∈ [0 , π )), these relations canbe summarized as h ˆ M i f = (cid:16) cos α Re h ˆ A i w + sin α Im h ˆ A i w (cid:17) θ + O ( θ ) . (4)Now let us examine the change in the probe wavepacket width. The variance of ˆ X for | ˜ φ f i is calculated (a) (b) FIG. 2. Wigner functions of the normalized probe statesafter the post-selection, | ˜ φ f i / k| ˜ φ f ik . We assume h ˆ A i w = 0and neglect O ( θ ) in Eq. (3). The horizontal and vertical axescorrespond to the observables ˆ X and ˆ K , respectively, and the45 ◦ and 135 ◦ axes, the observables ˆ Ω and ˆ Ξ , respectively. (a)When Re σ ( ˆ A ) θ = 0 . σ ( ˆ A ) θ = 0, the wave packetspreads along the X axis, while it narrows along the K axis.(b) When Re σ ( ˆ A ) θ = 0 and Im σ ( ˆ A ) θ = 0 .
5, the wavepacket spreads along the Ω axis, while narrows along the Ξ axis. as σ ( ˆ X ) = h ˆ X i f − h ˆ X i = σ ( ˆ X ) + 12 Re (cid:16) h ˆ A i w − h ˆ A i (cid:17) θ + O ( θ ) . (5)The real part of the variance-like quantity appears in thequadratic term of θ , which is ignored in the conventionalweak measurement context. We define this quantity as weak variance σ ( ˆ A ) for ˆ A : σ ( ˆ A ) := h ˆ A i w − h ˆ A i . (6)While the real part of the weak variance is similar to thenormal variance in that it appears in the change in theprobe wave packet width in ˆ X basis as in Eq. (2), it canbe negative unlike the normal variance; in that case, thewave packet width decreases as shown in Fig. 1(e). Thedecrease of the probe wave packet width reported in theprevious studies [22, 23] can also be understood as theeffect of the negative weak variance.We next consider where the imaginary part of the weakvariance appears. The variance of the generalized posi-tion operator ˆ M for | ˜ φ f i is calculated as [42] σ ( ˆ M ) = 12 + 12 h cos(2 α )Re σ ( ˆ A )+ sin(2 α )Im σ ( ˆ A ) i θ + O ( θ ) . (7)This equation indicates that the real and imaginaryparts of the weak variance appear in the changes of theprobe wave packet width in different measurement bases.For example, when choosing ˆ M = ˆ K ( α = π/ K for | ˜ φ f i is given as σ ( ˆ K ) = σ ( ˆ K ) − (1 / σ ( ˆ A ) θ + O ( θ ); that is, Re σ ( ˆ A ) also appearsin the change of the wave packet width in ˆ K basis. Theserelations can be understood on the quadrature phaseplane as shown in Fig. 2(a). When Re σ ( ˆ A ) >
0, thewave packet spreads along the X axis (horizontal axis),while it narrows along the K axis (vertical axis). Thisrelationship satisfies Kennard-Robertson uncertainty re-lation [43, 44] up to the quadratic of θ : σ ( ˆ X ) σ ( ˆ K ) =1 / O ( θ ).On the other hand, when α = π/
4, the measured ob-servable becomes ˆ M = ( ˆ X + ˆ K ) / √ Ω , which is theobservable corresponding to the 45 ◦ axis in the quadra-ture phase plane of Fig. 2. From the relation σ ( ˆ Ω ) = σ ( ˆ Ω ) + (1 / σ ( ˆ A ) θ + O ( θ ), the imaginary partof the weak variance Im σ ( ˆ A ) can be observed from thechange of the probe wave packet width in ˆ Ω basis. More-over, when α = 3 π/
4, the measured observable becomesˆ M = ( − ˆ X + ˆ K ) / √ Ξ , which is the canonical conju-gate of ˆ Ω satisfying [ ˆ Ω, ˆ Ξ ] = iˆ1 and corresponds to the135 ◦ axis in the quadrature phase plane of Fig. 2. Fromthe relation σ ( ˆ Ξ ) = σ ( ˆ Ξ ) − (1 / σ ( ˆ A ) θ + O ( θ ),Im σ ( ˆ A ) also appears in the change of the wave packetwidth in ˆ Ξ basis. These relation can be understoodon the quadrature phase plane as shown in Fig. 2(b).When Im σ ( ˆ A ) >
0, the wave packet spreads along the Ω axis (45 ◦ axis), while it narrows along the Ξ axis(135 ◦ axis). This relationship also satisfies Kennard-Robertson uncertainty relation up to the quadratic of θ : σ ( ˆ Ω ) σ ( ˆ Ξ ) = 1 / O ( θ ). III. EXPERIMENTAL DEMONSTRATION OFOBSERVATION OF WEAK VARIANCES
To verify the existence of the effects of the weak vari-ance, we experimentally demonstrated the observationof the weak variance using the optical system shown inFig. 3. In this setup, the target and probe systems are po-larization and transverse spatial modes of the laser beam(Menlo Systems C-fiber 780, central wavelength 780 nm),respectively. The polarization mode is a two-state systemspanned by, e.g., horizontal and vertical polarization ba-sis {| H i , | V i} or diagonal (45 ◦ ) and anti-diagonal (135 ◦ )polarization basis {| D i := ( | H i + | V i ) / √ , | A i := ( | H i −| V i ) / √ } . The pre- and postselection {| i i , | f i} in thepolarization mode was prepared using Glan–Thompsonprisms (GTPs), a half-wave plate (HWP), and quarter-wave plates (QWPs). The initial transverse distributionof the beam’s amplitude was prepared in a Gaussian dis-tribution φ ( X ) = π − / exp( − X / X is the Laser Preselection PostselectionWeak interaction
FIG. 3. Experimental setup for the observation of weak vari-ances. GTP: Glan–Thompson prism, HWP: half-wave plate,QWP: quarter-wave plate, SP: Savart plate, CCD: charge-coupled device. (a) Experimental setup when the probe sys-tem is measured in ˆ X basis. In the preselection, a HWP andtwo QWPs are used to prepare the weak variance to be (i)negative real and (ii) positive pure imaginary, respectively.(b)–(d) Experimental setups when the probe system is mea-sured in ˆ Ω , ˆ K , and ˆ Ξ basis, respectively. The lens (focallength f = 1 m) and free space propagation perform a frac-tional Fourier transform on the transverse distribution of thebeam. dimensionless position variable normalized by the stan-dard deviation of this distribution. The weak interactionexp( − i θ ˆ A ⊗ ˆ K ) was implemented using a Savart plate(SP), which is composed of two orthogonal birefringentcrystals ( β -BaB O , 1 mm thickness). In our setup, ˆ A was chosen as ˆ A = | D ih D | − | A ih A | and then SP causesa transverse shift of the diagonally (anti-diagonally) po-larized beam by a distance θ ( − θ ). The probe systemwas finally measured in ˆ X , ˆ Ω , ˆ K , and ˆ Ξ basis. ˆ X measurement for the transverse intensity distribution ofthe beam was implemented using charge-coupled device(CCD) camera (Teledyne Princeton Instruments ProEM-HS:512BX3), as shown in Fig. 3(a). On the other hand,the intensity measurement in ˆ Ω , ˆ K , and ˆ Ξ basis were im-plemented by fractional Fourier transforming (for detail,see Appendix B) the beam distribution using a lens (fo-cal length f = 1 m) before ˆ X measurement by the CCDcamera, as shown in Figs. 3(b)–(d).In order to verify the effects of the real and imagi-nary parts of weak variance independently, we chose thepre- and postselected states {| i i , | f i} of the polarizationso that the weak variance becomes (i) negative real or (ii)positive pure imaginary. In the case (i), the preselectedstate | i i was prepared by passing the vertically polarizedbeam through HWP whose fast axis is rotated from the FIG. 4. Measurement results of ∆ h ˆ X i in the case (i). Thesolid blue line is the theoretical curve fitted to the measureddata. The red dashed line shows the weak value Re h ˆ A i w θ ( θ =0 . ϑ H . (Insets) Intensity distribution of thebeam for each ϑ H . When ϑ H is close to zero, the O ( θ ) termin Eq. (4) becomes dominant and the distribution becomesdifferent from Gaussian. vertical direction by the angle ϑ H ; the output state be-comes | i i = cos(2 ϑ H − π/ | D i + sin(2 ϑ H − π/ | A i . Thepostselected state was fixed to | f i = | H i . The weak valueand weak variance becomes real numbers as follows: h ˆ A i w = cos(2 ϑ H )sin(2 ϑ H ) , σ ( ˆ A ) = − cos(4 ϑ H )sin (2 ϑ H ) . (8)In the case (ii), the preselected state | i i was preparedby passing the vertically polarized beam through QWP1whose fast axis is rotated from the vertical directionby the angle ϑ Q and QWP2 whose fast axis is fixedin vertical direction; the output state becomes | i i =cos( ϑ Q − π/ | D i + e − i2 ϑ Q sin( ϑ Q − π/ | A i . The post-selected state was fixed to | f i = | H i , as in the case (i).The weak variance becomes a pure imaginary number asfollows: σ ( ˆ A ) = 2i cos(2 ϑ Q )sin (2 ϑ Q ) . (9)First, we observed the weak value that appears in thetransverse displacement of the beam’s intensity distri-bution. Figure 4 shows the measurement results of thedisplacement of the mean value of the beam’s intensitydistribution in the ˆ X basis for various ϑ H , ∆ h ˆ X i := h ˆ X i f − h ˆ X i i , in the case (i). When ϑ H is small, the pre-and postselected states are close to orthogonal and ∆ h ˆ X i becomes large. The blue solid line is the theoretical curveof ∆ h ˆ X i fitted to the measured values with the visibil-ity V and the coupling strength θ as fitting parameters(for detail, see Appendix C), from which V = 0 .
999 and θ = 0 .
032 were decided. The red dashed line is the theo-
FIG. 5. Measurement results of ∆ σ ( ˆ M ) in the cases that(i) the weak variance is negative real, and (ii) pure imagi-nary. (a)–(d) Results when the measurement bases are ˆ X ,ˆ Ω , ˆ K , and ˆ Ξ . The solid blue lines are the theoretical curvesfitted to the measured data. The red dashed lines are thetheoretical curves of the weak variance [cos(2 α )Re σ ( ˆ A ) +sin(2 α )Im σ ( ˆ A )] θ ( θ = 0 . ϑ Q . retical curve of the weak value Re h ˆ A i w θ ( θ = 0 . ϑ H . Most of the measured values are consistentwith this curve, which indicates that this measurementsof the weak values were valid.Next, we observed the weak variance that appears inthe change of the variance of the beam’s intensity dis-tribution. Here ∆ σ ( ˆ M ) := [ σ ( ˆ M ) − σ ( ˆ M )] /σ ( ˆ M )denotes the rate of change of the variance of the beam’sintensity distribution from the initial value in the basisof ˆ M = ˆ X cos α + ˆ K sin α . Figure 5 shows the mea-surement results of ∆ σ ( ˆ M ) ( ˆ M = ˆ X, ˆ Ω, ˆ K, ˆ Ξ ) for var-ious ϑ Q , in each of cases (i) and (ii). When ϑ Q issmall, the pre- and postselected states are close to or-thogonal and large variance changes were observed. Theblue solid lines are the theoretical curves of ∆ σ ( ˆ M )fitted to the measured values. While the fixed value θ = 0 .
032 was used, the visibility V was used as a fit-ting parameter, considering the difference in alignment in P r o b a b ili t y Observable's eigenvalue O b s e r v a b l e ' s e i g e n v a l u e W ea k - v a l u e d p r o b a b ili t y (a) Pre-selected system (b) Pre- and post-selected system FIG. 6. (a) Probability distribution of the projective measure-ment of ˆ A for a preselected system. The expectation value andvariance of this distribution are h ˆ A i and σ ( ˆ A ), respectively.(b) Weak-valued probability distribution of the observable ˆ A for a pre- and postselected systems. The weak-valued proba-bilities p w j are complex and expressed on the complex plane. h ˆ A i w and σ ( ˆ A ) are also complex, but cannot be depicted inthis graph. each measurement (for detail, see Appendix C). The reddashed lines are the theoretical curves of the weak vari-ance [cos(2 α )Re σ ( ˆ A ) + sin(2 α )Im σ ( ˆ A )] θ ( θ = 0 . ϑ Q and most of the measured values are con-sistent with these curves. It should be noted that, inthe case (i), ∆ σ ( ˆ X ) certainly shows negative due tothe effect of the negative real weak variance; correspond-ingly, ∆ σ ( ˆ K ) increases. On the other hand, since theimaginary part of the weak variance is zero, ∆ σ ( ˆ Ω ) and∆ σ ( ˆ Ξ ) remain almost zero. In the case (ii), since theweak variance is positive pure imaginary, ∆ σ ( ˆ Ω ) and∆ σ ( ˆ Ξ ) show positive and negative, respectively. Onthe other hand, since the real part of the weak varianceis zero, ∆ σ ( ˆ X ) and ∆ σ ( ˆ K ) remain almost zero. Theseresults indicates that it was observed the real and imag-inary parts of the weak variance appear in the change ofthe wave packet width according to our theory. IV. STATISTICAL INTERPRETATION OFWEAK VARIANCE AS STATISTIC OFWEAK-VALUED PROBABILITY DISTRIBUTION
Here we describe that the weak variance is expressedas a statistic of the weak-valued probability distribution,which is a pseudo-probability distribution for the pre-and post-selected system. This relation is similar to thatthe variance is expressed as a statistic of the probabilitydistribution for the preselected system. This statisticalinterpretation of the weak variance, together with theoperational interpretation described above, rationalizesthe definition of the weak variance as a counterpart ofthe variance for the pre- and postselected systems.The weak-valued probability p w j := h ˆ Π j i w is definedas the weak value of each element of the set of projectionoperators { ˆ Π j } j that satisfy the completeness condition P j ˆ Π j = ˆ1. Weak-valued probabilities can be any com-plex number outside [0 , j is unity: P j p w j = 1. By regarding the weak-valued probabilityas a quantity corresponding to the probability that thepre- and post-selected particle is found in the eigenspaceˆ Π j between the pre- and postselection, a probabilisticinterpretation approach has been given to fundamentalproblems in quantum mechanics [4–10]. In addition, thenegativity and non-reality of the weak-valued probabili-ties have played an essential role in the studies such asthe investigation of the relationship between disturbanceand complementarity in quantum measurement [12, 13],the explanation of the violation of Bell inequality usingnegative probabilities [11], quantum enhancement of thephase estimation sensitivity by postselection [24], under-standing of out-of-time-order correlators as witness forquantum scrambling [25, 26].The weak value h ˆ A i w and weak variance σ ( ˆ A ) of theobservable ˆ A = P j a j ˆ Π j can be expressed as follows us-ing the weak-valued probabilities { p w j } j : h ˆ A i w = X j a j p w j (10) σ ( ˆ A ) = X j ( a j − h ˆ A i w ) p w j = X j | a j − h ˆ A i w | p w j , (11)where the second equation in Eq. (11) holds when ˆ A isHermite. These expressions are similar to those of theexpectation value h ˆ A i = P j a j p j and variance σ ( ˆ A ) = P j ( a j − h ˆ A i ) p j using the probability distribution { p j } j ,respectively. In this sense, the weak value and the weakvariance can be regarded as the expectation value and thevariance for the weak-valued probability distribution, re-spectively. In addition, since the weak-valued probabil-ity is represented as the conditional pseudo-probabilityfor the Kirkwood-Dirac distribution [45, 46], the weakvalue and weak variance are also regarded as the con-ditional pseudo-expectation value and the conditionalpseudo-variance for the Kirkwood-Dirac distribution, re-spectively (for detail, see Appendix D). Furthermore, theweak value and the weak variance satisfy the equationssimilar to the law of total expectation and the law oftotal variance, respectively: h ˆ A i = h i | ˆ A | i i = X j |h f j | i i| h ˆ A i w j , (12) σ ( ˆ A ) = X j |h f j | i i| σ j ( ˆ A ) + X j |h f j | i i| (cid:16) h ˆ A i w j − h ˆ A i (cid:17) , (13)where h ˆ A i w j := h f j | ˆ A | i i / h f j | i i and σ j ( ˆ A ) := h ˆ A i w j − h ˆ A i j So far, several definitions of the quantity correspondingto the variance for the pre- and postselected systems havebeen considered, such as the form of the weak varianceintroduced in Eq. (6) [27–31], its absolute value [32, 33],its real part [34, 39], and other forms [35–38]. Our dis-cussion in this paper have manifested that our form ofthe weak variance has similarities to the normal vari-ance in terms of both the measurement method usingindirect measurement and the statistical expression us-ing the weak-valued probability distribution. Therefore,the weak variance defined in Eq. (6) can be regarded rea-sonable as a counterpart of the variance for the pre- andpost-selection system.
V. CONCLUSION
In this paper, we have introduced the weak variance σ ( ˆ A ) as a complex counterpart of the variance for pre-and postselected systems. We have theoretically de-scribed that the weak variance appears in the change ofthe probe wave packet width in indirect measurementfor pre- and postselected systems, and experimentallydemonstrated it using an optical setup. We have alsodescribe that the weak value h ˆ A i w and the weak variance σ ( ˆ A ) are expressed as the statistics of the weak-valuedprobability distribution { p w j } j . These operational andstatistical interpretations are similar to the expectationvalue h ˆ A i and the variance σ ( ˆ A ) for preselected systems.Therefore, our formulation of the weak variance can beconsidered to be a reasonable definition as a counterpartof the variance for the pre- and postselected systems.The rest of this paper further extends the concept ofthe weak variance. We define n -th order weak moment ofthe observable ˆ A as h ˆ A n i w . The set of the weak moments {h ˆ A k i w } nk =1 fully characterizes the weak-valued proba-bility distribution { p w j } nj =1 , which is similar to the rela-tion between the set of the moments {h ˆ A k i} nk =1 and theprobability distribution { p j } nj =1 . The n -th order weakmoment h ˆ A n i w can be experimentally observed in theindirect measurement setup by taking into considerationthe terms up to the n -th order of θ in Eq. (3). The phys-ical meaning of the weak moment may be somewhat elu-sive as well as the weak values; nevertheless, the weakmoment could be possible to provide a new perspec-tive on fundamental problems in quantum mechanics.For example, Scully et al. ’s claim that the momentumdisturbance associated with which-way measurement inYoung’s double-slit experiment can be avoided [47] hasbeen justified by the fact that the weak-valued probabil-ities corresponding to the momentum disturbance havezero variance due to their negativeness [12, 13]. These studies are implicitly based on the idea of the weak vari-ance (second-order weak moment). In a similar way, theweak moment is expected to play an important role inother problems of this kind. In addition, the use of mea-surement methods other than the weak measurementswith Gaussian probes—such as weak measurement us-ing a qubit probe [48] and the method without a probe[49]—may find new implications for the weak moments.Finally, as an application of the weak moments h ˆ A n i w ,we propose the control of the probe wave packet by pre-and postselection of the target system. So far, severalstudies have been reported to narrow the probe wavepacket by appropriately pre- and postselecting the targetsystem in the weak measurement setup [22, 23, 50]. InEq. (3), if the higher-order weak moments included inthe O ( θ ) term are properly controlled, any waveform ofthe probe can be configured (for detail, see Appendix E).For example, the non-Gaussian states in the quadratureamplitude of light, such as the cat state [51, 52] and theGottesman–Kitaev–Preskill state [53], play an importantrole in quantum optics, and our method may provide anew construction method for their realization. ACKNOWLEDGMENTS
This research was supported by JSPS KAKENHIGrant Number 16K17524 and 19K14606, the MatsuoFoundation, and the Research Foundation for Opto-Science and Technology.
Appendix A: Change of probe wave packet in indirect measurement for mixed pre- and postselected systems
We calculate the expectation value h ˆ M i f and the variance σ ( ˆ M ) of the probe wave packet in the indirect mea-surement, when the pre- and postselected states are mixed states represented by the density operators ˆ ρ i and ˆ ρ f ,respectively. When ˆ ρ i = | i ih i | and ˆ ρ f = | f ih f | , it corresponds to the case of the pure pre- and postselected states, andwhen ˆ ρ f = ˆ1 /d , the case of the preselection only. The time evolution of the whole state is calculated as:ˆ ρ i ⊗ | φ ih φ | Interaction −−−−−−−→ exp( − i θ ˆ A ⊗ ˆ K ) (ˆ ρ i ⊗ | φ ih φ | ) exp(i θ ˆ A ⊗ ˆ K ) Postselection −−−−−−−−→ tr t h ˆ ρ f exp( − i θ ˆ A ⊗ ˆ K ) (ˆ ρ i ⊗ | φ ih φ | ) exp(i θ ˆ A ⊗ ˆ K ) i = tr (ˆ ρ f ˆ ρ i ) (cid:20) | φ ih φ | + (cid:18) − i θ h ˆ A i w ˆ K | φ ih φ | − θ h ˆ A i w ˆ K | φ ih φ | (cid:19) + H . c . + θ ˜ A ˆ K | φ ih φ | ˆ K (cid:21) + O ( θ )=: ˜ˆ ρ φ , (A1)where tr t denotes the partial trace for the target system, and H . c . represents the Hermitian conjugate of the precedingterm. h ˆ A i w = tr(ˆ ρ f ˆ A ˆ ρ i ) / tr(ˆ ρ f ˆ ρ i ) is the weak value of ˆ A for the pre- and postselected system represented by the densityoperators ˆ ρ i and ˆ ρ f . We define ˜ A := tr(ˆ ρ f ˆ A ˆ ρ i ˆ A ) / tr (ˆ ρ f ˆ ρ i ); when ˆ ρ i and ˆ ρ f are pure, ˜ A = |h ˆ A i w | , and when ˆ ρ f = ˆ1 /d (completely mixed state), ˜ A = tr(ˆ ρ i ˆ A ) = h ˆ A i .The expectation value of ˆ M for the unnormalized probe state ˜ˆ ρ φ is expressed as h ˆ M i f = tr(˜ˆ ρ φ ˆ M ) / tr(˜ˆ ρ φ ). Thenumerator tr(˜ˆ ρ φ ˆ M ) is calculated astr(˜ˆ ρ φ ˆ M ) = tr (ˆ ρ f ˆ ρ i ) (cid:20) h ˆ M i + (cid:18) − i θ h ˆ A i w h ˆ M ˆ K i − θ h ˆ A i w h ˆ M ˆ K i (cid:19) + c . c . + θ ˜ A h ˆ K ˆ M ˆ K i (cid:21) + O ( θ ) , (A2)where c . c . represents the complex conjugate of the preceding term. Because the expectation value of the product ofodd numbers of ˆ X or ˆ K for our Gaussian probe state becomes zero, for ˆ M = ˆ X cos α + ˆ K sin α , the above equationcan be reduced as tr(˜ˆ ρ φ ˆ M ) = tr (ˆ ρ f ˆ ρ i ) h(cid:16) − i θ h ˆ A i w h ˆ M ˆ K i (cid:17) + c . c . i + O ( θ ) . (A3)A similar calculation yields the denominator tr(˜ˆ ρ φ ) astr(˜ˆ ρ φ ) = tr (ˆ ρ f ˆ ρ i ) (cid:20) (cid:18) − θ h ˆ A i w h ˆ K i (cid:19) + c . c . + θ ˜ A h ˆ K i (cid:21) + O ( θ ) . (A4)Therefore, the expectation value h ˆ M i f is expressed as h ˆ M i f = tr(˜ˆ ρ φ ˆ M )tr(˜ˆ ρ φ ) = (cid:16) − i θ h ˆ A i w h ˆ M ˆ K i (cid:17) + c . c . + O ( θ ) , (A5)where we used the following formula: a + a θ + a θ + O ( θ ) b + b θ + b θ + O ( θ ) = a b + a b − a b b θ + a b − a b b − a b b + a b b θ + O ( θ ) . (A6)Because h ˆ M ˆ K i = (i cos α + sin α ) / h ˆ M i f as follows: h ˆ M i f = 12 θ h ˆ A i w (cos α − i sin α ) + c . c . + O ( θ ) = θ (cid:16) Re h ˆ A i w cos α + Im h ˆ A i w sin α (cid:17) + O ( θ ) , (A7)which matches Eq. (4) in the main text.The variance of ˆ M for the unnormalized probe state ˜ˆ ρ φ is expressed as σ ( ˆ M ) = h ˆ M i f − h ˆ M i . The first term iscalculated as h ˆ M i f = tr(˜ˆ ρ φ ˆ M )tr(˜ˆ ρ φ )= h ˆ M i + (cid:16) − θ h ˆ A i w h ˆ M ˆ K i (cid:17) + c . c . + θ ˜ A h ˆ K ˆ M ˆ K i + O ( θ )1 + (cid:16) − θ h ˆ A i w h ˆ K i (cid:17) + c . c . + θ ˜ A h ˆ K i + O ( θ )= h ˆ M i + (cid:18) − θ h ˆ A i w h ˆ M ˆ K i (cid:19) + c . c . + θ ˜ A h ˆ K ˆ M ˆ K i + h ˆ M i (cid:18) θ h ˆ A i w h ˆ K i + c . c . − θ ˜ A h ˆ K i (cid:19) + O ( θ ) . (A8)Because the following equations hold for our Gaussian probe state: h ˆ M i = h ˆ K i = 12 , h ˆ M ˆ K i = 14 −
12 [cos(2 α ) − i sin(2 α )] , and h ˆ K ˆ M ˆ K i = 34 , (A9)we obtain the following expression: h ˆ M i f = tr(˜ˆ ρ φ ˆ M )tr(˜ˆ ρ φ ) = 12 + 12 θ cos(2 α )Re h ˆ A i w + 12 θ sin(2 α )Im h ˆ A i w + 12 θ ˜ A + O ( θ ) . (A10)The second term h ˆ M i is calculated by using Eq. (A7) as h ˆ M i = 12 θ (cid:20)(cid:16) Re h ˆ A i w (cid:17) − (cid:16) Im h ˆ A i w (cid:17) (cid:21) cos(2 α ) + θ Re h ˆ A i w Im h ˆ A i w sin(2 α ) + 12 θ |h ˆ A i w | + O ( θ ) . (A11)Therefore, we obtain the concrete form of σ ( ˆ M ) as σ ( ˆ M ) = h ˆ M i f − h ˆ M i = 12 + 12 θ cos(2 α ) (cid:20) Re h ˆ A i w − (cid:16) Re h ˆ A i w (cid:17) + (cid:16) Im h ˆ A i w (cid:17) (cid:21) + 12 θ sin(2 α ) h Im h ˆ A i w − h ˆ A i w Im h ˆ A i w i + 12 θ (cid:16) ˜ A − |h ˆ A i w | (cid:17) + O ( θ )= 12 + 12 θ cos(2 α )Re σ ( ˆ A ) + 12 θ sin(2 α )Im σ ( ˆ A ) + 12 θ (cid:16) ˜ A − |h ˆ A i w | (cid:17) + O ( θ ) , (A12)where we used the following formulae:Re σ ( ˆ A ) = Re h ˆ A i w − (cid:16) Re h ˆ A i w (cid:17) + (cid:16) Im h ˆ A i w (cid:17) , (A13)Im σ ( ˆ A ) = Im h ˆ A i w − h ˆ A i w Im h ˆ A i w . (A14)In particular, when ˆ ρ i and ˆ ρ f are pure, ˜ A = |h ˆ A i w | and thus Eq. (A12) matches Eq. (7) in the main text. On theother hand, when ˆ ρ f = ˆ1 /d (case of preselection only), ˜ A = h ˆ A i , h ˆ A i w = h ˆ A i , and σ ( ˆ A ) = σ ( ˆ A ) ∈ R ; therefore, σ ( ˆ M ) = 12 + 12 θ cos(2 α ) σ ( ˆ A ) + 12 θ σ ( ˆ A ) + O ( θ ) . (A15)When α = 0, we obtain σ ( ˆ M ) = σ ( ˆ X ) = 12 + θ σ ( ˆ A ) + O ( θ ) , (A16)which is equal to σ ( ˆ X ) in Eq. (2) in the main text except for the O ( θ ) term (this term vanishes in the full-orderexpansion in this case). We note that if the probe state is not the Gaussian wave packet, the expectation value andvariance of ˆ M for the probe wave packet after the postselection will not be Eqs. (A7) and (A12), respectively.0 Appendix B: Fractional Fourier transform and its optical realization1. Definition of fractional Fourier transform
For any real number α , the α -angle fractional Fourier transform of a function φ is defined by F α [ φ ]( ω ) := r − i cot( α )2 π Z ∞−∞ d xφ ( x ) exp (cid:20) i (cid:18) cot( α ) ω − α ) ωx + cot( α ) x (cid:19)(cid:21) , (B1)where x and ω are dimensionless variables, cot( α ) = 1 / tan( α ), and csc( α ) = 1 / sin( α ). We also define φ α as thefunction into which the function φ is transformed by F α . In particular, when α = π/ F α is reduced to the normalFourier transform F : F π/ [ φ ]( ω ) = φ π/ ( ω ) = 1 √ π Z ∞−∞ d xφ ( x )e − i ωx = F [ φ ]( ω ) . (B2)When α = ± π/ α = ± π/
4, the fractional Fourier transform is respectively expressed as F ± π/ [ φ ]( ω ) = φ ± π/ ( ω ) = r ∓ i2 π Z ∞−∞ d xφ ( x ) exp " ± i ω − √ ωx + x ! , (B3) F ± π/ [ φ ]( ω ) = φ ± π/ ( ω ) = r ± i2 π Z ∞−∞ d xφ ( x ) exp " ∓ i ω + 2 √ ωx + x ! . (B4)We call F ± π/ and F ± π/ as ± /
2- and ± /
2. Relationship between observables ˆ X , ˆ K , ˆ Ω , and ˆ Ξ For an observable ˆ X , its canonical conjugate observable ˆ K is defined as the observable that satisfies the canonicalcommutation relation [ ˆ X, ˆ K ] = iˆ1. ˆ X and ˆ K are spectrally decomposed as follows:ˆ X = Z ∞−∞ d XX | X ih X | , ˆ K = Z ∞−∞ d KK | K ih K | . (B5)Their eigenvectors | X i and | K i are related to each other by the Fourier transform: | K i = F − π/ [ | X i ]( K ) = 1 √ π Z ∞−∞ d X | X i e i KX . (B6)ˆ Ω and ˆ Ξ are defined as ˆ Ω := ˆ X + ˆ K √ , ˆ Ξ := − ˆ X + ˆ K √ . (B7)They satisfy the canonical commutation relation [ ˆ Ω, ˆ Ξ ] = iˆ1. ˆ Ω and ˆ Ξ spectrally decomposed as follows:ˆ Ω = Z ∞−∞ d ΩΩ | Ω ih Ω | , ˆ Ξ = Z ∞−∞ d ΞΞ | Ξ ih Ξ | . (B8)1Their eigenvector | Ω i and | Ξ i are related to | X i by the − /
2- and − / | Ω i = F − π/ [ | X i ]( Ω ) = r π Z ∞−∞ d X | X i exp " − i Ω − √ ΩX + X ! , (B9) | Ξ i = F − π/ [ | X i ]( Ξ ) = r − i2 π Z ∞−∞ d X | X i exp " i Ξ + 2 √ ΞX + X ! . (B10)When the state | φ i is expanded in each basis as | φ i = R ∞−∞ d Xφ ( X ) | X i = R ∞−∞ d Ωφ π/ ( Ω ) | Ω i = R ∞−∞ d Kφ π/ ( K ) | K i = R ∞−∞ d Ξφ π/ ( Ξ ) | Ξ i , the relation between each wavefunction and basis vector is summarizedas follows: φ ( X ) F π/ −−−→ φ π/ ( Ω ) F π/ −−−→ φ π/ ( K ) F π/ −−−→ φ π/ ( Ξ ) , (B11) | X i F − π/ −−−−→ | Ω i F − π/ −−−−→ | K i F − π/ −−−−→ | Ξ i . (B12)
3. Optical realization of measurement of observables ˆ X , ˆ K , ˆ Ω , and ˆ Ξ We describe how to optically realize the measurement of observables ˆ X , ˆ K , ˆ Ω , and ˆ Ξ for the photon beam withtransverse distribution state | φ i . The measurement of ˆ X , which is the dimensionless transverse position observable,can be realized by measuring the photon’s transverse position using a photon detector with spatial resolution. Themeasurement of ˆ K can be realized by optically Fourier-transforming the photon’s wavefunction φ ( X ) = h X | φ i into φ π/ ( K ) and measuring its transverse position. The optical Fourier transform is realized by the combination of lenspassage and free-space propagation. In a similar manner, the measurement of ˆ Ω and ˆ Ξ can be realized by optically 1 / / φ ( X ) into φ π/ ( Ω ) and φ π/ ( Ξ ), respectively, and measuring its transverse position.In the following, we describe how to realize the optical 1 /
2- and 3 / z direction. x , k , and k x are the transverse position, the totalwavenumber, and the x component of the wavevector, respectively. We apply the paraxial approximation and assumethat k does not depend on k x because k x ≪ k . We define the dimensionless variables X := xk and K x := k x /k . Thefree-space propagation by the distance d is represented in the wavenumber space by the following transfer function: H free D ( K x ) ∝ exp (cid:18) − i DK x (cid:19) , (B13)where D := dk is a dimensionless distance. In the position space, this free-space propagation is also represented bythe convolution with the following function: h free D ( X ) ∝ Z ∞−∞ d K x H free D ( K x )e i K x X ∝ exp (cid:18) i X D (cid:19) . (B14)On the other hand, passing through a lens with focal length f is represented in the position space by the followingtransfer function: h lens F ( X ) ∝ exp (cid:18) − i X F (cid:19) , (B15)where F := f k is a dimensionless focal length. In the wavevector space, passing through this lens is also representedby the convolution with the following function: H lens F ( K x ) ∝ Z ∞−∞ d Xh lens F ( X )e − i K x X ∝ exp (cid:18) i F K x (cid:19) . (B16)If a photon with a transverse wavefunction φ ( X ) passes through a lens with focal length f , propagates in free space by2distance d , and passes through another lens with focal length f in that order, the resultant wavefunction is calculatedas follows: φ ( X ) lens f −−−−→ φ ( X ) exp (cid:18) − i X F (cid:19) (B17) free-space propagation d −−−−−−−−−−−−−−−→ Z ∞−∞ d X ′ φ ( X ′ ) exp (cid:18) − i X ′ F (cid:19) exp (cid:20) i ( X − X ′ ) D (cid:21) (B18) lens f −−−−→ Z ∞−∞ d X ′ φ ( X ′ ) exp (cid:18) − i X ′ F (cid:19) exp (cid:20) i ( X − X ′ ) D (cid:21) exp (cid:18) − i X F (cid:19) = Z ∞−∞ d X ′ φ ( X ′ ) exp (cid:20) i ( F − D ) X − F XX ′ + ( F − D ) X ′ F D (cid:21) . (B19)If we choose D = F , we can realize the normal Fourier transform of φ as follows:Eq. ( B
19) = Z ∞−∞ d X ′ φ ( X ′ ) exp (cid:18) i − XX ′ F (cid:19) ∝ F π/ [ φ ] (cid:18) XF (cid:19) = φ π/ (cid:18) XF (cid:19) . (B20)The scale of the wavefunction after the Fourier transform can be adjusted by the focal length F . If we choose D = (1 − / √ F ,Eq. ( B
19) = Z ∞−∞ d X ′ φ ( X ′ ) exp " i X − √ XX ′ + X ′ (2 − √ F = Z ∞−∞ d X ′ φ ( X ′ ) exp i2 X q ( √ − F − √ X q ( √ − F X ′ q ( √ − F + X ′ q ( √ − F ∝ F π/ h φ − ,F i X q ( √ − F (cid:20) φ − ,F ( X ) := φ (cid:18) X q ( √ − F (cid:19)(cid:21) = φ − π/ ,F X q ( √ − F , (B21)where φ − ,F ( X ) is a scaled wavefunction of φ ( X ). In this manner, we can realize the 1 / φ F, ( X ).Similarly, if we choose D = (1 + 1 / √ F , we can realize the 3 / B
19) = Z ∞−∞ d X ′ φ ( X ′ ) exp " − i X + 2 √ XX ′ + X ′ (2 + √ F = Z ∞−∞ d X ′ φ ( X ′ ) exp − i2 X q ( √ F + 2 √ X q ( √ F X ′ q ( √ F + X ′ q ( √ F ∝ F π/ h φ +0 ,F i X q ( √ F (cid:20) φ +0 ,F ( X ) := φ (cid:18) X q ( √ F (cid:19)(cid:21) = φ − π/ ,F X q ( √ F , (B22)where φ +0 ,F ( X ) is a scaled wavefunction of φ ( X ).It should be noted that the second lens, which causes phase modulation in the position space, does not affect theresult of intensity (projection) measurement in the position basis. Therefore, in our experiment in the main text, the3intensity measurements of the beam’s transverse distribution in ˆ X , ˆ Ω , ˆ K and ˆ Ξ bases have been realized by usingonly one lens followed by free-space propagation. Appendix C: Derivation of theoretical values of expectation value and variance of probe system in ourexperiment
First, we derive the exact formula of the weak value and the weak variance in our experimental setup. Accordingto our experiment, we assume the pre- and postselected states {| i i , | f i} and the observable ˆ A as | i i = cos ϑ i | i + e i ϕ i sin ϑ i | i , | f i = 1 √ | i + | i ) , ˆ A = | ih | − | ih | . (C1)The weak value and the weak variance are calculated as h ˆ A i w = cos ϑ i + i sin ϑ i sin ϕ i ϑ i cos ϕ i , (C2) σ ( ˆ A ) = 2 sin ϑ i (sin ϑ i + cos ϕ i ) − i sin(2 ϑ i ) sin ϕ i (1 + sin ϑ i cos ϕ i ) . (C3)In our experiment, we used the following values: ϑ i = ( ϑ H − π/ ϑ Q − π/ , ϕ i = ( − ϑ Q [case (ii)] . (C4)By substituting them for Eqs. (C2) and (C3), we obtain Eqs. (8) and (9).Next, we derive the theoretical values of the expectation value and the variance of the probe wave packet in ourexperiment. The state of the whole system after the interaction isexp( − i θ ˆ A ⊗ ˆ K ) | i i| φ i = cos ϑ i | i exp( − i θ ˆ K ) | φ i + e i ϕ i sin ϑ i | i exp(i θ ˆ K ) | φ i . (C5)Suppose that the first and second terms on the right-hand side of Eq. (C5) are denoted by | Φ i and | Φ i , respectively.Considering the decrease in visibility V ∈ [0 , ρ := | Φ ih Φ | + | Φ ih Φ | + V ( | Φ ih Φ | + | Φ ih Φ | ) . (C6)After the target system is postselected onto | f i , the unnormalized probe state is given as ˜ˆ ρ f := h f | ˆ ρ | f i . We assumethat the initial probe state is Gaussian distribution h X | φ i = φ ( X ) = π − / exp( − X / M = ˆ X cos α + ˆ K sin α for ˜ˆ ρ f is calculated as h ˆ M i f = tr(˜ˆ ρ f ˆ M )tr(˜ˆ ρ f ) = θ cos α cos ϑ i + V sin α sin ϑ i sin ϕ i e − θ V sin ϑ i cos ϕ i e − θ . (C7)The theoretical values of the expectation value in the case (i) is obtained by substituting Eq. (C4) as h ˆ M i f = θ cos α sin(4 ϑ H )1 − V cos(4 ϑ H )e − θ . (C8)We fitted this function to the measured data with V and θ as the fitting parameters, and then obtained V = 0 . θ = 0 . M for ˜ˆ ρ f is calculated as σ ( ˆ M ) = h ˆ M i f − h ˆ M i = 12 + θ sin ϑ i n(cid:16) cos α − V sin α e − θ (cid:17) sin ϑ i + V e − θ [cos(2 α ) cos ϕ i + sin(2 α ) cos ϑ i sin ϕ i ] o(cid:0) V sin ϑ i cos ϕ i e − θ (cid:1) . (C9)Substituting Eq. (C4), the theoretical values of the variance in the case (i) is obtained as σ ( ˆ M ) = 12 + θ cos(4 ϑ H ) h(cid:16) cos α − V e − θ sin α (cid:17) cos(4 ϑ H ) − V e − θ cos(2 α ) i(cid:2) − V cos(4 ϑ H )e − θ (cid:3) , (C10)and that in case (ii) as σ ( ˆ M ) = 12 + θ cos(2 ϑ Q ) n(cid:16) cos α − V e − θ sin α (cid:17) cos(2 ϑ Q ) − V e − θ (cid:2) cos(2 α ) cos(2 ϑ Q ) − sin(2 α ) sin (2 ϑ Q ) (cid:3)o(cid:2) − V cos (2 ϑ Q )e − θ (cid:3) . (C11)We fitted these functions to the measured data with V as the fitting parameter and θ = 0 .
032 as the fixed parameter.
Appendix D: Weak variance as conditional pseudo-variance of Kirkwood–Dirac distribution
We show that weak values and weak variances can be understood as conditional pseudo-expectation values andconditional pseudo-variances of Kirkwood–Dirac (KD) distribution [45, 46], respectively. The ( j, k ) component of theKD distribution of the state | i i expanded in two orthonormal bases {| a j i} j and {| a ′ k i} k is defined as D ( a j , a ′ k | i) := tr( | a j ih a j | a ′ k ih a ′ k | i ih i | ) . (D1)The KD distribution is a joint pseudo-probability distribution that represents the quantum state | i i and is generallya complex number. The sum of the KD distribution for the two indices j, k becomes unity: P jk D ( a j , a ′ k | i) = 1. Themarginal distribution of KD distribution summed for one index becomes the projection probability distribution of | i i in the other basis: X j D ( a j , a ′ k | i) = |h a ′ k | i i| , X k D ( a j , a ′ k | i) = |h a j | i i| . (D2)When we choose | a j i = | f i , the conditional pseudo-probability D ( a ′ k | i , f) of the KD distribution is expressed as: D ( a ′ k | i , f) := D (f , a ′ k | i) P k D (f , a ′ k | i) = h f | a ′ k ih a ′ k | i ih f | i i = p ′ w k , (D3)where { p ′ w k } k is the weak-valued probability distribution of the pre- and postselection system {| i i , | f i} in the orthonor-mal basis {| a ′ k i} k . Therefore, the weak value h ˆ A ′ i w and weak variance σ ( ˆ A ′ ) for the observable ˆ A ′ := P k a ′ k | a ′ k ih a ′ k | are represented as the conditional pseudo-expectation values and conditional pseudo-variance of the KD distribution,respectively, as follows: X k a ′ k D ( a ′ k | i , f) = X k a ′ k p ′ w k = h ˆ A ′ i w , (D4) X k ( a ′ k − h ˆ A ′ i w ) D ( a ′ k | i , f) = X k ( a ′ k − h ˆ A ′ i w ) p ′ w k = σ ( ˆ A ′ ) . (D5)5 Appendix E: Control of probe wavefunction by pre- and postselection of target system
We explain that, in indirect measurement for pre- and postselected systems, the probe state after the postselectioncan be controlled by appropriately choosing the pre- and postselection. The wavefunction in the ˆ K basis of the probestate after the postselection | ˜ φ i [Eq. (3) in the main text], ˜ φ ( K ), is expressed for all orders of θ as˜ φ ( K ) = h K | ˜ φ i = h K | " h f | i i ∞ X n =0 ( − i θ ) n n ! h ˆ A n i w ˆ K n | φ i = h f | i i ∞ X n =0 ( − i θ ) n n ! h ˆ A n i w K n φ ( K ) . (E1)Let φ ⋆ ( K ) be the wavefunction in the ˆ K basis of the desired probe state. To realize φ ⋆ ( K ) except for a constantmultiple, we can choose the weak moments {h ˆ A n i} n so that ∞ X n =0 ( − i θ ) n n ! h ˆ A n i w K n ∝ φ ⋆ ( K ) φ ( K ) . (E2)When the target system is d -dimensional, if ˆ A has full rank, we can choose the values of d weak moments h ˆ A n i independently. Therefore, by fixing the values of the weak moments of the low-order term of θ ( n = 1 , · · · , d )appropriately, which has a large effect on the waveform, the desired wavefunction can be approximately realized. [1] J. A. Wheeler and W. H. Zurek, Quantum theory andmeasurement (Princeton University Press, 2014).[2] J. von Neumann,
Mathematical foundations of quan-tum mechanics: New edition (Princeton university press,2018).[3] Y. Aharonov, D. Z. Albert, and L. Vaidman, “Howthe result of a measurement of a component of thespin of a spin- particle can turn out to be 100,”Phys. Rev. Lett. , 1351 (1988).[4] Y. Aharonov and L. Vaidman, “Complete description ofa quantum system at a given time,” J. Phys. A , 2315(1991).[5] K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Ex-perimental realization of the quantum box problem,”Phys. Lett. A , 125 (2004).[6] J. S. Lundeen and A. M. Steinberg, “Experimental jointweak measurement on a photon pair as a probe of hardy’sparadox,” Phys. Rev. Lett. , 020404 (2009).[7] K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Di-rect observation of hardy’s paradox by joint weak mea-surement with an entangled photon pair,” New J. Phys. , 033011 (2009).[8] T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel,A. Matzkin, J. Tollaksen, and Y. Hasegawa, “Obser-vation of a quantum cheshire cat in a matter-wave inter-ferometer experiment,” Nat. Commun. , 5492 (2014).[9] R. Okamoto and S. Takeuchi, “Experimental demonstra-tion of a quantum shutter closing two slits simultane-ously,” Sci. Rep. , 35161 (2016).[10] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman,“Asking photons where they have been,” Phys. Rev. Lett. , 240402 (2013).[11] B. L. Higgins, M. S. Palsson, G. Y. Xiang, H. M. Wise-man, and G. J. Pryde, “Using weak values to experimen-tally determine “negative probabilities” in a two-photon state with bell correlations,” Phys. Rev. A , 012113(2015).[12] H. M. Wiseman, “Directly observing momentum trans-fer in twin-slit “which-way” experiments,” Phys. Lett. A , 285–291 (2003).[13] R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Stein-berg, J. L. Garretson, and H. M. Wiseman, “A double-slit ‘which-way’ experiment on the complementarity–uncertainty debate,” New J. Phys. , 287 (2007).[14] Y. Xiao, H. M. Wiseman, J.-S. Xu, Y. Kedem, C.-F.Li, and G.-C. Guo, “Observing momentum disturbancein double-slit “which-way” measurements,” Sci. Adv. ,eaav9547 (2019).[15] M. Ringbauer, D. N. Biggerstaff, M. A. Broome,A. Fedrizzi, C. Branciard, and A. G. White, “Experi-mental joint quantum measurements with minimum un-certainty,” Phys. Rev. Lett. , 020401 (2014).[16] F. Kaneda, S.-Y. Baek, M. Ozawa, and K. Edamatsu,“Experimental test of error-disturbance uncertainty re-lations by weak measurement,” Phys. Rev. Lett. ,020402 (2014).[17] L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat,Y. Soudagar, and A. M. Steinberg, “Violation of heisen-berg’s measurement-disturbance relationship by weakmeasurements,” Phys. Rev. Lett. , 100404 (2012).[18] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P.Mirin, L. K. Shalm, and A. M. Steinberg, “Observingthe average trajectories of single photons in a two-slitinterferometer,” Science , 1170 (2011).[19] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J.Resch, H. M. Wiseman, and A. M. Steinberg, “Ex-perimental nonlocal and surreal bohmian trajectories,”Sci. Adv. , e1501466 (2016).[20] J. Dressel, C. J. Broadbent, J. C. Howell, andA. N. Jordan, “Experimental violation of two-party leggett-garg inequalities with semiweak measurements,”Phys. Rev. Lett. , 040402 (2011).[21] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon,J. L. O’Brien, A. G. White, and G. J. Pryde, “Violationof the leggett–garg inequality with weak measurementsof photons,” PNAS , 1256 (2011).[22] B. de Lima Bernardo, W. S. Martins, S. Azevedo, andA. Rosas, “Uncertainty control and precision enhance-ment of weak measurements in the quadratic regime,”Phys. Rev. A , 012109 (2015).[23] F. Matsuoka, A. Tomita, and Y. Shikano, “Generation ofphase-squeezed optical pulses with large coherent ampli-tudes by post-selection of single photon and weak cross-kerr non-linearity,” Quantum Stud.: Math. Found. ,159–169 (2017).[24] D. R. M. Arvidsson-Shukur, N. Y. Halpern, H. V. Lep-age, A. A. Lasek, C. H. W. Barnes, and S. Lloyd,“Quantum negativity provides advantage in postselectedmetrology,” arXiv preprint arXiv:1903.02563.[25] N. Yunger Halpern, B. Swingle, and J. Dressel,“Quasiprobability behind the out-of-time-ordered corre-lator,” Phys. Rev. A , 042105 (2018).[26] J. R. Gonz´alez Alonso, N. Yunger Halpern, andJ. Dressel, “Out-of-time-ordered-correlator quasiproba-bilities robustly witness scrambling,” Phys. Rev. Lett. , 040404 (2019).[27] B. Reznik and Y. Aharonov, “Time-symmetric formula-tion of quantum mechanics,” Phys. Rev. A , 2538–2550(1995).[28] B. Reznik, “Interaction with a pre and post selectedenvironment and recoherence,” arXiv preprint quant-ph/9501023 (1995).[29] A. Tanaka, “Semiclassical theory of weak values,”Phys. Lett. A , 307–312 (2002).[30] A. Brodutch, “Weak measurements of non local vari-ables,” arXiv preprint arXiv:0811.1706 (2008).[31] A. D. Parks, “Weak energy: form and function,” in Quan-tum Theory: A Two-Time Success Story (Springer, 2014)pp. 291–302.[32] Y. Aharonov and L. Vaidman, “Properties of a quantumsystem during the time interval between two measure-ments,” Phys. Rev. A , 11–20 (1990).[33] A. D. Parks, “Weak covariance and the correlation of anobservable with pre-selected and post-selected state ener-gies during its time-dependent weak value measurement,”Quantum Stud.: Math. Found. , 455–461 (2018).[34] M. R. Feyereisen, “How the weak variance of momentumcan turn out to be negative,” Found. Phys. , 535–556(2015).[35] H. F. Hofmann, “Characterization of decoherence in a quantum channel using weak measurements,” in Interna-tional Quantum Electronics Conference (Optical Societyof America, 2011) p. I260.[36] H. F. Hofmann, “On the role of complex phases in thequantum statistics of weak measurements,” New J. Phys. , 103009 (2011).[37] A. K. Pati and J. Wu, “Uncertainty and complemen-tarity relations in weak measurement,” arXiv preprintarXiv:1411.7218 (2014).[38] Q.-C. Song and C.-F. Qiao, “Uncertainty equalitiesand uncertainty relation in weak measurement,” arXivpreprint arXiv:1505.02233 (2015).[39] P. P. Hofer, “Quasi-probability distributions for observ-ables in dynamic systems,” Quantum , 32 (2017).[40] The dimensionless variable X is obtained by dividing theposition variable x by the standard deviation σ of thewave function π − / σ − / exp[ − x / (2 σ )].[41] R. Jozsa, “Complex weak values in quantum measure-ment,” Phys. Rev. A , 044103 (2007).[42] See Appendix A for the detail when the pre- and posts-elected states of the target system are mixed states.[43] E. H. Kennard, “Zur quantenmechanik einfacher bewe-gungstypen,” Z. Phys. , 326–352 (1927).[44] H. P. Robertson, “The uncertainty principle,” Phys. Rev. , 163–164 (1929).[45] J. G. Kirkwood, “Quantum statistics of almost classicalassemblies,” Phys. Rev. , 31–37 (1933).[46] P. A. M. Dirac, “On the analogy between classical andquantum mechanics,” Rev. Mod. Phys. , 195–199(1945).[47] M. O. Scully, B.-G. Englert, and H. Walther, “Quantumoptical tests of complementarity,” Nature , 111–116(1991).[48] S-.J. Wu and K. Mølmer, “Weak measurements with aqubit meter,” Phys. Lett. A , 34–39 (2009).[49] K. Ogawa, H. Kobayashi, and A. Tomita, “Opera-tional formulation of weak values without probe sys-tems,” Phys. Rev. A , 042117 (2020).[50] A. D. Parks and J. E. Gray, “Variance control in weak-value measurement pointers,” Phys. Rev. A , 012116(2011).[51] B. Yurke and D. Stoler, “Generating quantum mechan-ical superpositions of macroscopically distinguishablestates via amplitude dispersion,” Phys. Rev. Lett. ,13–16 (1986).[52] W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassicalstate from two pseudoclassical states,” Phys. Rev. A ,2172–2187 (1991).[53] D. Gottesman, A. Kitaev, and J. Preskill, “Encoding aqubit in an oscillator,” Phys. Rev. A64