Complex counterpart of variance in quantum measurements for pre- and post-selected systems

The variance of an observable in a pre-selected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and post-selected systems, we formulate a complex-valued counterpart of the variance called"weak variance."In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and post-selected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and post-selected systems.


I. INTRODUCTION
The outcomes of quantum measurements show probabilistic behavior. This characteristic, which is not observed in classical systems, has been the root of many fundamental arguments in quantum theory [1]. In the quantum measurement of an observableÂ, the probabilistic behavior of its measurement outcomes is characterized by measurement statistics such as expectation value Â and variance σ 2 (Â). These values are generally measured using an indirect measurement method [2]. In indirect measurement, the target system to be measured is coupled with an external probe system through von Neumann interaction. Regardless of the coupling strength, the expectation value Â and variance σ 2 (Â) in the target system are obtained from the displacement of the probe wave packet and the increase in its width, respectively. In other words, the probe wave packet in an indirect measurement serves as the interface that displays the probabilistic characteristics of the target system.
Interestingly, when the target system is further postselected, the displacement of the probe wave packet differs from Â . In particular, when the coupling strength is weak (weak measurement setup), the probe displacement is given by Re Â w , where Â w := f|Â|i / f|i in the pre-and post-selected system {|i , |f } is called weak value [3]. Â w is complex in general and can exceed the spectral range ofÂ. By regarding the weak value as a complex counterpart of the expectation value * ogawak@ist.hokudai.ac.jp in the pre-and post-selected system, new approaches to fundamental problems in quantum mechanics involving pre-and post-selection have been investigated, such as various quantum paradoxes [4][5][6][7][8][9][10], understanding of the violation of Bell's inequality using negative probabilities [11], the relationship between disturbance and complementarity in quantum measurements [12][13][14], verification of the uncertainty relations [15][16][17], observation of Bohmian trajectories [18,19], and demonstration of the violation of macrorealism [20,21].
Similar to the relation between the weak value and expectation value, does there exist a counterpart of the variance in pre-and post-selected systems? To answer this question, we consider the function of the probe wave packet in indirect measurement as an interface that displays the characteristics of the target system. As mentioned earlier, the variance σ 2 (Â) in a pre-selected system manifests as an increase in the probe wave packet width in indirect measurement, and owing to the non-negativity of the variance, the wave packet width never decreases. However, for pre-and post-selected systems, any counterparts of the variance cannot be observed in the typical framework of the weak measurement [3], in which the probe wave packet width does not change because the second-and higher-order terms of the coupling strength are ignored. Here, we focus on the recent studies reporting that when considering the second-and higher-order terms of the coupling strength, the probe wave packet width can not only increase but also decrease under appropriate pre-and post-selection conditions [22,23]. If these reported phenomena are interpreted to result from a counterpart of the variance in pre-and post-selected systems, it may be possible to formulate an effective variance-like quantity that can be negative.
In this study, we investigate the general changes in the width of the probe wave packet during indirect measurements of pre-and post-selected systems. We then formulate a counterpart of the variance in these systems. This counterpart, denoted here as weak variance, can indeed be negative and manifests as the decrease in the probe wave packet width. Moreover, the weak variance is generally complex and can be understood by observing the changes in the probe wave packet width on the quadrature phase plane. To demonstrate this phenomenon, we conducted an optical experiment for observing the changes in the beam packet width in proportion to the real and imaginary parts of the complex weak variance. In addition, to clarify the concept of weak variance, we express the weak variance as the second-order moment of the weak-valued probability distribution [4][5][6][7][8][9][10][11][12][13][24][25][26], which is a quasi-probability distribution in pre-and post-selected systems. Based on the agreement between the operational and statistical interpretations, we propose that our weak variance can be considered a reasonable definition of a complex counterpart of the variance in pre-and post-selected systems than previous formulations [27][28][29][30][31][32][33][34][35][36][37][38][39]. Furthermore, we formulate a counterpart of the higher-order moment and investigate its operational and statistical meanings and applications.

II. WEAK VARIANCE APPEARING IN INDIRECT MEASUREMENT FOR PRE-AND POST-SELECTED SYSTEMS
The indirect measurements have been made with a Gaussian probe as shown in Figs. 1(a) and (d). After reviewing these measurements, we explain the appearance of a complex weak variance in the pre-and postselected system. The target system to be measured and the probe system are pre-selected in states |i and |φ , respectively. The initial probe state |φ can be expanded as |φ = ∞ −∞ dXφ(X)|X , where the wave function φ(X) is the Gaussian distribution φ(X) = π −1/4 exp(−X 2 /2) and X is a dimensionless variable [40]. The observable of the dimensionless positionX can be spectrally decomposed asX = ∞ −∞ dXX|X X|. The time evolution by the interaction HamiltonianÂ ⊗K is represented by the unitary operatorÛ (θ) = exp(−iθÂ ⊗K), wherê A = j a jΠj is the observable to be measured in the target system, a j is an eigenvalue ofÂ,Π j is the pro-  . The distribution of probe wave packets after the interaction reproduces the probability distribution of the outcomes of projective measurements ofÂ in |i . (c) Change in the probe wave packet under the weak coupling condition (θ Â ≪ 1), where the horizontal axis has been rescaled from that of (b). The variance of the probe wave packet after the interaction increases in proportion to the variance σ 2 (Â) and never decreases. (d) Quantum circuit of indirect measurements of the pre-and post-selected system {|i , |f } (weak measurement setup). (e) Change in the probe wave packet in the weak measurement circuit (d). The real part of the weak variance appears in the variance change of the probe wave packet after the post-selection. Unlike the pre-selected system (c), the pre-and post-selected system admits a narrowed variance of the probe wave packet when the real part of the weak variance becomes negative.
jector onto the eigenspace ofÂ belonging to eigenvalue a j ,K is the canonical conjugate observable ofX satisfying [X,K] = i1, and θ is a parameter with the reciprocal dimension ofÂ. The coupling strength is characterized by θ Â , where Â is the largest eigenvalue of A: if θ Â ≫ 1 (≪ 1), the coupling is considered strong (weak).
Let us consider an indirect measurement of the observ-ableÂ, as shown in Fig. 1 (a). Suppose that measure-mentX in the probe system is made to the state after the interaction, |Ψ := exp(−iθÂ ⊗K)|i |φ . The probability distribution P (X) of obtaining the result X is where p j := i|Π j |i is the projection probability of |i ontoΠ j . If the coupling is strong (θ Â ≫ 1), the wave packet |φ(X − θa j )| 2 for each j is well separated from other wave packets, and P (X) reproduces the probability distribution {p j } j [ Fig. 1(b)]. However, if the coupling is weak (θ Â ≪ 1), the wave packets overlap and P (X) does not reproduce {p j } j [ Fig. 1(c)]. Nevertheless, regardless of the coupling strength, the statistics ofÂ in the target system |i , such as the expectation value Â and the variance σ 2 (Â), can be acquired from the changes in the probe distribution P (X). The expectation value and variance of X in P (X) are respectively expressed as where X i and σ 2 i (X) are the expectation value and variance ofX in the initial probe state |φ , respectively. In this case, X i = 0 and σ 2 i (X) = 1/2. Therefore, the expectation value Â and variance σ 2 (Â) can be measured under both strong and weak coupling conditions. Here, we stress that after the interaction, the variance of the probe wave packet σ 2 f (X) never decreases because the variance σ 2 (Â) is non-negative.
We next consider that the target system is pre-and post-selected in states |i and |f , respectively [ Fig. 1(d)]. The non-normalized state of the probe system after the post-selection |φ f := f|Ψ is represented as The expectation value ofX in the non-normalized state The real part of the weak value Â w = f|Â|i / f|i appears in the displacement of the probe wave packet, as previously reported for weak measurements [3]. The imaginary part of the weak value is observed in the displacement of the probe wave packet in theK basis: K f = Im Â w θ + O(θ 3 ) [41]. By introducing the generalized position operatorM :=X cos α +K sin α (α ∈ [0, 2π)), these relations can be summarized as Now let us examine the change in the probe wave packet width. The variance ofX for |φ f is calculated as The real part of the variance-like quantity appears in the quadratic term of θ, which is ignored in the conventional weak measurement context. We define this quantity as the weak variance σ 2 w (Â) ofÂ: The real part of the weak variance is similar to normal variance in that it appears as a change in the probe wave packet width in theX basis [Eq. (2)]. However, unlike the normal variance, the weak variance can be negative, in which case the wave packet width then decreases as shown in Fig. 1(e). The decrease in the probe wave packet width reported in previous studies [22,23] can be reinterpreted as the effect of the negative weak variance.
We next consider the appearance of the imaginary part of the weak variance. The variance of the generalized position operatorM for |φ f is calculated as (see Appendix A for a detailed analysis of mixed pre-and postselected states of the target system) This equation indicates that the real and imaginary parts of the weak variance appear in the changes in the probe wave packet width in different measurement bases. For example, when choosingM =K (α = π/2), the variance ofK in state |φ f is given as ; that is, Reσ 2 w (Â) also appears in the width change of the wave packet in theK basis. To clarify these relations, we plot them on the quadrature phase plane [ Fig. 2(a)]. When Reσ 2 w (Â) > 0, the wave packet spreads along the X (horizontal) axis, while it narrows along the K (vertical) axis. This relationship satisfies the Kennard-Robertson uncertainty relation [42,43] up to the quadratic of θ: . However, when α = π/4, the measured observable be-comesM = (X +K)/ √ 2 =:Ω, which corresponds to the 45 • axis in the quadrature phase plane of Fig. 2 , the imaginary part of the weak variance Imσ 2 w (Â) can be observed as the change in the probe wave packet width in theΩ basis. Moreover, when α = 3π/4, the measured observable becomesM = (−X +K)/ √ 2 =:Ξ, which is the canonical conjugate ofΩ. This observable satisfies [Ω,Ξ] = i1 and corresponds to the 135 • axis in the quadrature phase plane of Fig. 2. Through the relation , Imσ 2 w (Â) also appears in the width change of the wave packet in theΞ basis. These relations are represented on the quadrature phase plane in Fig. 2(b). When Imσ 2 w (Â) > 0, the wave packet spreads along the Ω (45 • ) axis, while it narrows along the Ξ (135 • ) axis. This relationship also satisfies the Kennard-Robertson uncertainty relation up to the quadratic of θ:

III. EXPERIMENTAL DEMONSTRATION OF WEAK VARIANCES
To verify the effects of the weak variance, we experimentally observed the weak variance in the optical system shown in Fig. 3. In this setup, the target and probe systems were the polarization and transverse spa- The pre-and post-selection {|i , |f } in the polarization mode was prepared using Glan-Thompson prisms (GTPs), a half-wave plate (HWP), and quarterwave plates (QWPs). The initial transverse distribution of the beam's amplitude was prepared as a Gaussian distribution φ(X) = π −1/4 exp(−X 2 /2), where X is the dimensionless position variable normalized by the standard deviation of this distribution. The weak interaction exp(−iθÂ ⊗K) was implemented using a Savart plate (SP), which comprises two orthogonal birefringent crystals (β-BaB 2 O 4 , 1-mm thickness). In our setup,Â was chosen asÂ = |D D| − |A A| and the SP transversely shifted the diagonally (antidiagonally) polarized beam by a distance of θ (−θ). The probe system was finally measured in theX,Ω,K, andΞ bases. In theX basis, the transverse intensity distribution of the beam was measured using a charge-coupled device (CCD) camera (Teledyne Princeton Instruments ProEM-HS:512BX3), as shown in Fig. 3(a). The intensity measurements in theΩ,K, andΞ bases were implemented by fractional Fourier transforming (see Appendix B for details) the beam distribution using a lens (focal length f = 1 m) be-foreX measurement using the CCD camera, as shown in To independently verify the effects of the real and imaginary parts of the weak variance, we chose the preand post-selected polarization states {|i , |f } giving (i) negative real and (ii) positive purely imaginary weak variances. The pre-selected state |i in case (i) was prepared by rotating the fast axis of the HWP through angle ϑ H from the vertical direction and passing the vertically polarized beam through the rotated HWP. The output state became |i = cos(2ϑ H −π/4)|D +sin(2ϑ H −π/4)|A . The post-selected state was fixed at |f = |H . The weak value and weak variance respectively became the following real numbers: The pre-selected state |i in case (ii) was prepared by rotating the fast axis of QWP1 through angle ϑ Q from the vertical direction and passing the vertically polarized beam through the rotated QWP1 and QWP2 (whose fast axis was fixed in the vertical direction). The output state became |i = cos(ϑ Q − π/4)|D + e −i2ϑQ sin(ϑ Q − π/4)|A . The post-selected state was fixed at |f = |H , as in case (i). The weak variance became the following purely imaginary number: First, we observed the weak value in the transverse displacement of the beam's intensity distribution. Figure 4 plots the measured displacement of the mean of the beam's intensity distribution in theX basis, given by ∆ X := X f − X i , as a function of ϑ H in case (i). When ϑ H is small, the pre-and post-selected states are nearly orthogonal and ∆ X becomes large. The theoretical curve of ∆ X (blue solid curve in Fig. 4) Figure 5 plots the measured ∆σ 2 (M ) (M =X,Ω,K,Ξ) as functions of ϑ H and ϑ Q in cases (i) and (ii), respectively. When ϑ H and ϑ Q were small, the pre-and post-selected states were close to orthogonal and the variance changes were large. The theoretical curves of ∆σ 2 (M ) (blue solid curves in Fig. 5) were fitted to the measured values using θ, V , ∆, and the intensity of the background light N as fitting parameters (see Appendix C for details). The theoretical weak variances cos(2α)Reσ 2 w (Â) + sin(2α)Imσ 2 w (Â) θ 2 (red-dashed curves) are also plotted as functions of ϑ H or ϑ Q in Fig. 5. Again, most of the measured values were consistent with the theoretical curves. In case (i), ∆σ 2 (X) became negative because the weak variance was a negative real value; correspondingly, the ∆σ 2 (K) increased. However, ∆σ 2 (Ω) and ∆σ 2 (Ξ) remained almost zero because the imaginary part of the weak variance was zero. In case (ii), where the weak variance was positive and purely imaginary, ∆σ 2 (Ω) and ∆σ 2 (Ξ) became positive and negative, respectively. However, ∆σ 2 (X) and ∆σ 2 (K) remained almost zero. Thus, the real and imaginary parts of the weak variance appeared as width changes of the wave packet, in accordance with our theory.

IV. WEAK VARIANCE AS A STATISTIC OF THE WEAK-VALUED PROBABILITY DISTRIBUTION
In this section, we interpret the weak variance as a statistic of the weak-valued probability distribution, which is a pseudo-probability distribution in the pre-and post-selected system. This relation is similar to that the variance is expressed as a statistic of the probability dis- tribution in the pre-selected system. This statistical interpretation of the weak variance, together with the operational interpretation described above, rationalizes the definition of the weak variance as a counterpart of the variance in pre-and post-selected systems.
We define the weak-valued probability p wj := Π j w as the weak value of each element of the set of projection operators {Π j } j that satisfy the completeness condition jΠ j =1. Weak-valued probabilities can be any complex number outside [0, 1], but their sum for all j is unity: j p wj = 1. By regarding the weak-valued probability as a quantity corresponding to the probability of finding a pre-and post-selected particle in the eigenspacê Π j between the pre-and post-selection, researchers have found a probabilistic approach to fundamental problems in quantum mechanics [4][5][6][7][8][9][10]. Because of their negativity and nonreality, weak-valued probabilities have played an essential role in studies such as the investigation of the relationship between disturbance and complementarity in quantum measurements [12,13], the explanation of the violation of Bell inequality using negative probabilities [11], quantum enhancement of the phase estimation sensitivity via post-selection [24], and understanding outof-time-order correlators as witnesses of quantum scrambling [25,26].
The weak value Â w and weak variance σ 2 w (Â) of ob-servableÂ = j a jΠj can be expressed in terms of the weak-valued probabilities {p wj } j as follows: where the second equation in Eq. (11) holds whenÂ is Hermitian. These expressions are similar to those of the expectation value Â = j a j p j and variance σ 2 (Â) = j (a j − Â ) 2 p j , respectively, obtained using probability distribution {p j } j . In this sense, the weak value and weak variance can be regarded as the expectation value and variance of a weak-valued probability distribution, respectively. In addition, as the weak-valued probability represents the conditional pseudo-probability of the Kirkwood-Dirac distribution [44,45], the weak value and weak variance are also regarded as the conditional pseudo-expectation value and conditional pseudovariance of the Kirkwood-Dirac distribution, respectively (see Appendix D for details). Furthermore, the weak value and weak variance satisfy the equations similar to the laws of total expectation and total variance, respectively: where Â wj := f j |Â|i / f j |i and σ 2 wj (Â) := Â 2 wj − Â 2 wj Thus far, several definitions of the quantity corresponding to the variance in pre-and post-selected systems have been considered. Examples are the weak variance expressed by Eq. (6) [27][28][29][30][31], its absolute value [32,33], its real part [34,39], and other forms [35][36][37][38]. The measurement method (indirect measurement) and statistical expression (a weak-valued probability distribution) of our proposed weak variance are similar to those of the conventional variance. Therefore, the weak variance defined by Eq. (6) can be regarded as a reasonable counterpart of the variance in pre-and post-selection systems.

V. CONCLUSION
We introduced the weak variance σ 2 w (Â) as a complex counterpart of the variance in pre-and post-selected systems. We theoretically showed that the weak variance appears as the changing width of the probe wave packet during indirect measurements of pre-and post-selected systems and experimentally demonstrated the weak variance in an optical setup. We also expressed the weak value Â w and weak variance σ 2 w (Â) as statistics of the weak-valued probability distribution {p wj } j . These operational and statistical interpretations are similar to the expectation value Â and variance σ 2 (Â) in pre-selected systems. Therefore, our formulation of the weak variance can be considered a reasonable definition of a counterpart of the variance in pre-and post-selected systems.
Extending the concept of the weak variance, we then defined the n-th order weak moment of the observablê A as Â n w . The set of weak moments { Â k w } n k=1 fully characterizes the weak-valued probability distribution {p wj } n j=1 . A similar relation exists between the set of moments { Â k } n k=1 and the probability distribution {p j } n j=1 . The n-th order weak moment Â n w can be experimentally observed in the indirect measurement setup by including terms up to the n-th order of θ in Eq. (3). Although the physical meanings of the weak moment is as elusive as the weak value, the weak moment may provide a new perspective on fundamental problems in quantum mechanics. For example, Scully et al.'s claim that the momentum disturbance associated with which-way measurement in Young's double-slit experiment can be avoided [46] has been justified by the negativity of the weak-valued probabilities corresponding to the momentum disturbance, which consequently have zero variance [12,13]. These studies are implicitly based on the weak variance (second-order weak moment) concept. Similarly, the weak moment is expected to play an important role in other problems of this type. In addition, measurement methods other than weak measurements with Gaussian probes-such as weak measurements using a qubit probe [47] and methods without a probe [48]-may find new implications for the weak moments.
Finally, as an application of the weak moments Â n w , we propose controlling the probe wave packet by preand post-selection of the target system. In several studies, the probe wave packet was narrowed by appropriate pre-and post-selection of the target system in the weak measurement setup [22,23,49]. If the higher-order weak moments in the O(θ 2 ) term of Eq. (3) are properly controlled, we can configure any waveform of the probe (see Appendix E for details). For example, our method may represent a new construction method for the realization of the non-Gaussian states in the quadrature amplitude of light, such as the cat state [50,51] and the Gottesman-Kitaev-Preskill state [52], which play important roles in quantum optics.
The expectation value ofM for the non-normalized probe stateρ φ is expressed as M f = tr(ρ φM )/tr(ρ φ ). The numerator tr(ρ φM ) is calculated as where c.c. represents the complex conjugate of the preceding term. Because the expectation value of the product of odd numbers ofX orK in our Gaussian probe state becomes zero, forM =X cos α +K sin α, the above equation can be reduced to By a similar calculation, the denominator tr(ρ φ ) is obtained as Therefore, the expectation value M f is expressed as where we have used the following formula: Because MK = (i cos α + sin α)/2 in our Gaussian probe state, we obtain a concrete form of M f as follows: which matches Eq. (4) in the main text.
The variance ofM for the non-normalized probe stateρ φ is expressed as σ 2 f (M ) = M 2 f − M 2 f . The first term is calculated as Because the following equations hold for our Gaussian probe state: we obtain the following expression: The second term M 2 f is calculated using Eq. (A7) as Therefore, we obtain the concrete form of σ 2 f (M ) as where we have used the following formulae: In particular, whenρ i andρ f are pure,Ã = | Â w | 2 , so Eq. (A12) matches Eq. (7) in the main text. However, when ρ f =1/d (pre-selection only),Ã = Â 2 , Â w = Â , and σ 2 w (Â) = σ 2 (Â) ∈ R; therefore, we have When α = 0, we obtain which equals σ 2 f (X) in Eq. (2) of the main text but without the O(θ 3 ) term (which vanishes in the full-order expansion in this case). Note that if the probe state is not a Gaussian wave packet, the expectation value and variance ofM for the probe wave packet after the post-selection will differ from Eqs. (A7) and (A12), respectively.
Appendix B: Fractional Fourier transform and its optical realization

Optical realization of measurement of observablesX,K,Ω, andΞ
This Appendix describes the optical system for measuring the observablesX,K,Ω, andΞ for a photon beam with a transverse distribution state |φ . To measure the dimensionless transverse-position observableX, we measure the photon's transverse position using a photon detector with suitable spatial resolution. To measureK, we optically Fourier-transform the photon's wavefunction φ 0 (X) = X|φ and measure the transverse position of the resulting function φ π/2 (K). The optical Fourier transform is realized by combining a lens passage with free-space propagation. Similarly,Ω andΞ can be measured by optically 1/2-and 3/2-Fourier-transforming φ 0 (X) and measuring the transverse positions of the resulting functions φ π/4 (Ω) and φ 3π/4 (Ξ), respectively. In what follows, we derive the optical 1/2-and 3/2-Fourier transform by combining the lens passage and free-space propagation.
We assume that the beam is propagating in the z direction and define x, k, and k x as the transverse position, total wavenumber, and x component of the wavevector, respectively. We apply the paraxial approximation and assume that k does not depend on k x because k x ≪ k. We then define the dimensionless variables X := xk and K x := k x /k. Free-space propagation through distance d is represented in wavenumber space by the following transfer function: where D := dk is the dimensionless distance. In position space, free-space propagation is represented by a convolution with the following function: Meanwhile, passage through a lens with focal length f is represented in the position space by the following transfer function: where F := f k is the dimensionless focal length. In wavevector space, passage through this lens is represented by a convolution with the following function: If a photon with a transverse wave function φ 0 (X) sequentially passes through a lens with focal length f , propagates in free space through distance d, and passes through another lens with focal length f , the resultant wave function is calculated as If we choose D = F , we obtain the standard Fourier transform of φ 0 : After the Fourier transform, the scale of the wave function can be adjusted by adjusting the focal length F . If we where φ − 0,F (X) is a scaled wave function of φ 0 (X). In this manner, we obtain the 1/2-Fourier transform of φ F,0 (X). Similarly, if we choose D = (1 + 1/ √ 2)F , the 3/2-Fourier transform is obtained as follows: where φ + 0,F (X) is a scaled wave function of φ 0 (X). Note that the second lens, which causes phase modulation in the position space, does not affect the measured intensity (projection) in the position basis. Therefore, in the experiment (see main text), the intensities of the beam's transverse distribution in theX,Ω,K andΞ bases were measured by inserting only one lens followed by free-space propagation.
Appendix C: Theoretical derivations of the expectation value and variance in our experimental probe system First, we derive the exact formulae of the weak value and weak variance in our experimental setup. In the experiment, we assumed the pre-selected state |i , post-selected state |f , and the observableÂ as The weak value and weak variance are calculated as In our experiment, we used the following values: Substituting these terms into Eqs. (C2) and (C3), we obtain Eqs. (8) and (9), respectively.
Next, we derive the theoretical curves of the expectation value and variance of the probe wave packet demonstrated in our experiment. The state of the entire system after the interaction is exp(−iθÂ ⊗K)|i |φ = cos ϑ i 2 |0 exp(−iθK)|φ + e iϕi sin ϑ i 2 |1 exp(iθK)|φ .
For notational simplicity, we denote the first and second terms on the right-hand side of Eq. (C5) by |Φ 0 and |Φ 1 , respectively. Considering the decrease in visibility V ∈ [0, 1], the state of the entire system after the interaction is expressed by the following density operator: After post-selecting the target system onto |f , the non-normalized probe state becomesρ f := f|ρ|f . The initial probe state is assumed as a Gaussian distribution X|φ = φ(X) = π −1/4 exp(−X 2 /2). The expectation value of the observableM =X cos α +K sin α forρ f is calculated as tr(ρ f ) = θ cos α cos ϑ i + V sin α sin ϑ i sin ϕ i e −θ 2 1 + V sin ϑ i cos ϕ i e −θ 2 . (C7) To obtain the theoretical curve of the expectation value ofX in case (i), we substitute Eq. (C4) and α = 0 into Eq. (C7) as X f = θ sin(4ϑ H ) 1 − V cos(4ϑ H )e −θ 2 . (C8) Here we assume a technical error in the rotation angle of the HWP ϑ H ∆ ∈ [−0.5 • , 0.5 • ] that occurs in the experiment; accordingly, the fitting function is given as Fitting this function to the measured data with θ, V and ∆ as the fitting parameters, we obtained θ = 3.62 × 10 −2 , V = 1.00, and ∆ = 1.57 × 10 −3 .
Meanwhile, the theoretical variance curve of the observableM forρ f , σ 2 f (M ), is calculated as follows. The nonnormalized probability density distribution of the signalρ in the measurement basisM = where L = 5.6 in our experimental setup. The normalized probability density distribution of the summed background and signal intensities is given by