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Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

What does it take for real-deterministic c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic c-valued variables out of the weak values obtained via a non-perturbing weak measurement of quantum operators with a post-selection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to transform the incompatibility between the quantum operator and the basis into the fluctuation of an `error term', and then superimpose it onto the real-part of the weak value. We show that this class of ``c-valued physical quantities'' provides a real-deterministic contextual hidden variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the c-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two c-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty relation is captured by a specific irreducible epistemic restriction, foreign in classical mechanics, constraining the allowed form of the joint distribution of the two c-valued physical quantities. We then suggest an epistemic interpretation of the two terms decomposing the c-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.


I. INTRODUCTION
Among the many profound and bizarre implications of the fundamental noncommutativity between the observables in quantum mechanics, the Heisenberg uncertainty principle stands to offer the most direct physical intuition about the quantum randomness [1]. The Kennard-Robertson-Schrödinger (KRS) uncertainty relation [2][3][4] -one of the manifestations of the quantum uncertainty principle -dictates that the product of the standard deviations of two observables is fundamentally bounded from below by a value determined by the commutation relation between the two observables. Since the beginning of quantum mechanics, some researchers have speculated that quantum randomness might be explained by deterministic variables associated with noncommuting quantum observables. The most famous argument in favour of such an approach is given by Einstein, Podolsky and Rosen in their seminal paper [5]. Bell [6,7] and Kochen and Specker [8] later showed, however, that, to comply with certain specific quantum predictions, such deterministic variables must violate some of the classically plausible assumptions about Nature. Here we ask: what classical assumptions we need to give in to construct real-valued deterministic variables associated with noncommuting quantum observables so that they respect the KRS uncertainty relation.
One way to study this problem is to construct such real-valued deterministic variables, and to identify the fundamental relation among them -foreign in classical mechanics -which capture the KRS uncertainty relation. Such a relation, if exists, cannot be seen directly via the standard strong (i.e., sharp) measurement in quantum mechanics since it does not allow a joint measurement of two noncommuting observables: a sharp measurement of one observable in general disturbs the system, making the observable noncommuting with the first becomes indefinite (indeterministic).
On the other hand, a novel scheme of measurement was introduced three decades ago by Aharonov, Albert and Vaidman which goes beyond the measurement-induced-disturbance in standard measurement [9,10]. In this scheme, given a pre-selected ensemble characterized by a quantum state |ψ , one first weakly measures a quantum observableÔ without significantly perturbing the quantum state. This is then followed with a post-selection over an ensemble characterized by |φ via a strong measurement which may be incompatible with the first weak measurement. Repeating the process over the pre-and post-selected ensembles, one gets a weak value: O w (φ|ψ) . = φ|Ô|ψ φ|ψ , where it is assumed that φ|ψ = 0. Using the measurement procedure, one may thus, to an extent, obtain simultaneous information about the two incompatible observables, otherwise inaccessible via the standard strong measurement. The real and imaginary parts of the weak value can be inferred respectively from the average shift of the position and momentum of the pointer of the measuring device [11,12]. The observation of the weak value via an arbitrarily nonperturbing weak measurement with postselection can also be argued to reflect our ordinary and naive idea of measurement in the presence of randomness [13]. Moreover, the fact that it is obtained via weak measurement intuitively suggests that it somehow reveals the counterfactual properties of the system compatible with pre-selection and post-selection [13][14][15][16][17][18][19][20][21]. This insight has motivated the application of the concept of weak value to study various quantum conundrum [14,15,19,[22][23][24][25][26][27][28][29][30][31][32], and the meaning of a quantum state [33][34][35]. We note that the real or/and imaginary parts of the weak value can also naturally arise in schemes not involving weak measurements [35][36][37][38][39][40][41][42], suggesting that it might have a deep nature beyond the operational context of weak measurement with post-selection. With all this in mind, it is interesting to ask to what extent the information contained in weak values is sufficient to construct deterministic variables associated with the noncommuting observables which capture the KRS uncertainty relation.
Indeed, some significant efforts have been made to reconstruct quantum statistics based on deterministic variables by using the weak values. It has been argued, for example, that weak values can capture the contextuality of the quantum fluctuations [43][44][45]. In particular, in Ref. [45], it is shown that the complex weak values explicitly reveal the contextuality of quantum fluctuations that are defined by the choice of the measurement post-selection and the quantum state preparation. Moreover, such deterministic weak values can also be used to derive the KRS uncertainty relation by subjecting them to classical uncertainty relation [46]. Note however that these weak values are in general complex even if the associated quantum operators are Hermitian. Such complex deterministic variables do not give a transparent insight into the fundamental relation among them which underlies the KRS uncertainty relation. Furthermore, it is desirable to have real-valued deterministic variables which not only smoothly generalize the strong values given by the real eigenvalues of Hermitian observables, but which also offer insight into a conceptually smooth quantumclassical correspondence.
In the present work, we discuss the above problems by devising a class of deterministic real-valued variables constructed from complex weak values which always satisfy the KRS uncertainty relation. This work essentially extends the results reported in Refs. [47,48] to finite dimensional discrete variable quantum systems. First, in Sec. II A, given a quantum operator, a quantum state associated with a preparation, and a complete set of state vectors in the Hilbert space as a reference coordinate basis, we construct a real-deterministic cvalued (i.e., classical, commuting) variable from the associated complex weak value with a post-selection over the basis, which we call 'c-valued physical quantity'. See Eq. (3). To do this, we introduce an auxiliary real-valued global-nonseparable (i.e., spatially uniform) random variable ξ and couple it to the imaginary part of the weak value, transforming the incompatibility between the quantum operator and the reference coordinate basis into the strength of the fluctuations of ξ. This ξ-dependent 'error term' is then superimposed onto a ξ-independent term given by the real part of the weak value. We show in Sec.
II B that this class of c-valued physical quantities provide a real-deterministic contextual hidden variable model for the quantum expectation value of a class of quantum operators.
The contexts underlying the c-valued physical quantities are given by the choice of the reference basis which defines the post-selection in the weak value measurement, the choice of weak measurement when the associated observable can be written into a product of two observables, and the choice of preparation associated with a given density matrix (mixed state).
In Sec. III, we show that varying the reference basis, the quantum variance of a Hermitian operator over a quantum state is decomposed into the trade-off between the strength of the ξ-independent and ξ-dependent fluctuations of the associated c-valued physical quantity.
The Schrödinger and the Kennard-Robertson lower bounds in quantum uncertainty relation can then be obtained separately by imposing the classical uncertainty relation respectively to the ξ-independent term and the ξ-dependent error term. In this representation, the finite Kennard-Robertson lower bound associated with two incompatible quantum observables reflects the absence of a common reference coordinate basis wherein the ξ−dependent error terms of the associated two c-valued physical quantities simultaneously vanish. We proceed in Sec. IV A to argue that the formalism captures the incompatibility between two observables in terms of a specific fundamental epistemic restriction between the associated two c-valued physical quantities. It constrains the allowed form of their joint distribution and thereby underlies the KRS uncertainty relation. In this sense, a specific form of epistemic freedom, fundamental in classical mechanics, has to be given in. We then suggest in Sec. IV B an epistemic interpretation of the two terms decomposing the c-valued physical quantity as the optimal estimate under epistemic restriction and the associated single-shot estimation error, and discuss the classical limit. Sec. V summarizes the present paper and offers a future outlook.

II. REAL-DETERMINISTIC C-VALUED PHYSICAL QUANTITIES
A. Real-deterministic physical quantities from complex weak values and an auxiliary global random variable First, given the Hilbert space associated with the quantum system under interest, let us choose a complete set of quantum states {|φ n } satisfying the completeness relation n |φ n φ n | =1, where the summation is replaced with a suitable integration for a continuous set, and1 is the identity matrix. We take this complete set of vectors in the Hilbert space as the reference coordinate basis. We then define a vector of weak values O w (φ n |ψ) associated with an operatorÔ with a pre-selected state |ψ and a post-selection over the complete set of quantum states {|φ n } as Let us denote the real and imaginary parts of O w (φ n |ψ) respectively as It should be noted importantly that, while the strong value refers to a single event (i.e., a single repetition of a strong measurement), the weak value is obtained by repeating the weak measurement over the pre-and post-selected ensembles and taking an average.
Next, for each quantum operatorÔ and a preparation characterized by a quantum state |ψ , let us define a real-deterministic c-valued physical quantityÕ(φ n , ξ|ψ) which depends on the choice of a reference coordinate basis {|φ n } as Here, ξ is a global-nonseparable variable assumed to fluctuate randomly on a microscopic time scale independent of the pre-selected state |ψ , the operatorÔ, and the reference coordinate basis {|φ n }. For example, we can assume that each single run of the weak measurement is assigned a single random realization of ξ. By global-nonseparable, we meant ξ is spatially uniform so that, two systems, arbitrarily spatially separated from each other, always experience the same simultaneous value of ξ. Moreover, we assume that its average and variance are respectively given by [47] ξ .
where χ(ξ) is the probability density of ξ. We shall here on refer to the ξ-dependent term on the right-hand side of Eq. (3) as the 'error term', suggesting that it is an epistemic rather than an ontic quantity. The reason for this naming will be made clear in Sec. IV B. An extension of the above defined c-valued physical quantity to a pre-selected mixed quantum state (density matrix) is given at the end of Sec. II B.
Operationally,Õ(φ n , ξ|ψ) defined in Eq. (3) can thus be obtained as follows. First, one makes weak measurement ofÔ with a pre-selected state |ψ and a post-selected state |φ n and infers the real and imaginary parts of the weak value from the average position and momentum of the pointer of the measuring device. Next, one generates ξ using a random number generator, i.e., sampling ξ from χ(ξ), satisfying Eq. (4), e.g., ξ = ± with half-half probability. Given the pair of (φ n , ξ), one then computesÕ(φ n , ξ|ψ) as prescribed in Eq.
Note that, for a given fixed value of φ n ,Õ(φ n , ξ|ψ) in general fluctuates randomly due to the fluctuations of ξ. AveragingÕ(φ n , ξ|ψ) over ξ, we obtain, by virtue of Eq. (4), the real part of the weak value, i.e.
Hence, O w R (φ n |ψ) can be seen as the conditional average ofÕ given φ n . Moreover, it is also clear from Eq. (2) that the fluctuations ofÕ(φ n , ξ|ψ) about its conditional average O w R (φ n |ψ), i.e., the fluctuations of the error term ξ O w I (φ n |ψ) in Eq. (3), becomes stronger as the incompatibility betweenÔ andΠ φn over |ψ is larger. An epistemic interpretation of the two decomposing terms on the right-hand side of Eq. (3) within an operational scheme of optimal estimation will be suggested in Sec. IV B.
For an arbitrary pre-selected state |ψ and a reference coordinate basis {|φ n }, we further assume that the coordinate value φ n is distributed according to the Born's rule. Namely, the probability that φ n is sampled is given by Pr(φ n |ψ) = φ n |ψ 2 . Moreover, we define the ensemble average of a c-valued function f (φ n , ξ|ψ) over the joint probability distribution Pr(φ n , ξ|ψ) = | φ n |ψ | 2 χ(ξ) of (φ n , ξ) as in the conventional probability theory, i.e., f (φ n , ξ|ψ) . = n dξf (φ n , ξ|ψ)χ(ξ) φ n |ψ 2 . It is then straightforward to show that the average ofÕ(φ n , ξ|ψ) defined in Eq. (3) over the distribution of (φ n , ξ) is given by the real part of the quantum expectation value ofÔ over |ψ , i.e., where we have used Eqs. (2) and (4), and the completeness of the basis {|φ n }. We also have, using again Eqs. (2), (3) and (4), From Eqs. (6) and (7), the expectation value of any quantum operatorÔ over an arbitrary pre-selected state |ψ can thus be decomposed as, Hence, it can be seen as the ensemble average of a complex c-valued variable defined as: As a comparison, the quantum expectation value ofÔ over |ψ can be expressed directly in terms of the average of weak value as ψ|Ô|ψ = n O w (φ n |ψ)| φ n |ψ | 2 . Note however that O w (φ n |ψ) is in general complex-valued even whenÔ is Hermitian. In contrast to this, in Hence, regardless the value of ξ, the c-valued physical quantity is equal to the real eigenvalue o n ofÔ, the standard strong value obtained in strong measurement. This is also the case when |ψ is given by one of the eigenstates ofÔ for arbitrary choices of basis. In this sense, O(φ n , ξ|ψ) may be regarded as a generalization of the strong value. Recall that strong value associated withÔ is definite only when the pre-selected state |ψ is given by one of the eigenstates ofÔ; otherwise it is indefinite when |ψ is a superposition of different eigenstates ofÔ. By contrast, the c-valued physical quantityÕ(φ n , ξ|ψ) is always definite for all |ψ .
Hence, by introducing ξ, we have somehow transformed the quantum coherence-induced indefiniteness in the strong value, into a fluctuation of a definite c-valued physical quantity.
Note further that even whenÔ is nonHermitian,Õ(φ n , ξ|ψ) defined in Eq. (3) is always real. However, in this case, even when |φ n or |ψ is the eigenstate ofÔ, since the associated eigenvalue may be complex, O w I (φ n |ψ) is in general not vanishing so thatÕ(φ n , ξ|ψ) may still fluctuate randomly due to the fluctuation of ξ. Hence, if we wishÕ(φ n , ξ|ψ) defined in Eq. (3) to be independent of ξ when |φ n or |ψ is the eigenstate ofÔ, thenÔ must be Hermitian.
Complementarity between the error terms. Consider two noncommuting quantum observables,Â andB, [Â,B] = 0, so that there is no common complete set of vectors in whichÂ andB are jointly diagonalized. Now, take the complete set of eigenvectors {|a n } ofÂ with the associated set of eigenvalues {a n } as the reference coordinate basis. As shown above, we thus haveÃ(a n , ξ|ψ) = a n , i.e., it is independent of the global random variable ξ. However, observe that, in this case, in general we have [Π an ,B] = 0 so that B w I = 0. Accordingly, B(a n , ξ|ψ) must in general fluctuate due to the fluctuation of ξ via the ξ-dependent error term. Moreover, this fluctuation is stronger as the incompatibility betweenB andΠ an over |ψ is larger. Hence, for an arbitrary pair of noncommuting quantum observables, if one chooses a reference coordinate basis so that the error term of the c-valued physical quantity associated with one of the observables is vanishing, the error term of the c-valued physical quantity associated with the other observable is in general not vanishing, and vice versa.
This complementarity between the error terms of the c-valued physical quantities associated with two noncommuting observables thus captures the complementarity between the randomness of the quantum strong values.
Hence, it in general does not conserve the functional relationship of the original operators.
The equality between the left and the right-hand sides of Eq. (11) applies when the preselected state |ψ or the post-selected state |φ n is the joint eigenstate of the two Hermitian operatorsÂ andB. As an illustrative example, consider the quantum Hamiltonian of a free particle with mass m in one spatial dimension:Ĥ =p 2 /2m. Taking the complete set of the eigenvectors {|q } of the position operatorq as the reference coordinate basis, one has where we have expressed the wave function in the polar form, i.e., ψ(q) . = q|ψ = ρ(q)e iS(q)/ , so that S(q) = Arg{ψ(q)}, and ρ(q) = |ψ(q)| 2 . In particular, while the right-hand side of Eq. (12) is always non-negative, the left-hand side may take negative value. As an example of such a negative c-valued kinetic energy, consider the case when the pre-selected state is given by a Gaussian wave function ψ(q) ∼ e −q 2 /4σ 2 q . Then, one has Interestingly, as will be shown in Sec. II B, the ensemble average of both sides in Eq. (11) are always equal. For weak values, the above feature for the product of operators is the key to many quantum paradoxes involving weak measurement with post-selection [10].

B. C-valued physical quantities as a real-deterministic contextual hidden variable model for a class of quantum expectation values
We first obtain the following mathematical result for the average of the product of two c-valued physical quantities. Consider an arbitrary pair of operatorsÂ andB. For any pre-selected state |ψ and a reference coordinate basis {|φ n }, the ensemble average of the product betweenÃ(φ n , ξ|ψ) associated withÂ, andB(φ n , ξ|ψ) associated withB, is equal to the quantum expectation value of the symmetrized product of the two operators. Namely, using Eqs. (3) and (4), we straightforwardly obtain Here, to get the second equality, we have used Eq. (4) and a simple mathematical identity Re{ab * } = Re{a}Re{b} + Im{a}Im{b} for two arbitrary complex numbers a and b. One can check that whenB =1, we regain Eq. (6). Note that while the decomposition in the second line of Eq. (13) depends on the choice of the coordinate basis {|φ n }, the last line (i.e., the right-hand side) is basis-independent.
Next, takingB =Â in Eq. (13), one obtains As an example of the above result, taking the ensemble average of both sides of Eq. (12), one has where the first and the last equalities are due to respectively Eqs. (9) and (16). Note that the equality between the first and the last ensemble averages in Eq. (17) can be checked directly by noting Eq. (4) and using the identity 1 4 ∂qρ ρ where ρ(q) = | q|ψ | 2 , and assuming that ρ(q) is vanishing at the boundary.
Observe that the cross terms, i.e., A w R(I) (φ n |ψ)B w I(R) (φ n |ψ), did not appear in the second line of Eq. (13) because of the vanishing average of ξ assumed in Eq. (4). Moreover, we emphasize that the globalness (i.e., the nonseparability) of ξ is indeed in general indispensable when the operatorsÂ andB act locally on two spatially separated systems. Otherwise, if ξ were separable into two independent random variables each belonging to the two systems, i.e., ξ = (ξ A , ξ B ) so that then, we have ξ A ξ B = ξ A × ξ B = 0 so that the second term in the second line of Eq. (13) vanishes. Further elaboration of this fact and its deep relation with quantum entanglement will be discussed somewhere else.
As a direct implication of Eq. (14), the quantum covariance of two Hermitian observableŝ A andB over a quantum state |ψ can be expressed as the classical covariance of the associated c-valued physical quantitiesÃ(φ n , ξ|ψ) andB(φ n , ξ|ψ), i.e., TakingÂ =B =Ô in Eq. (19), the quantum variance of an observableÔ over a quantum state |ψ can thus be expressed as the classical variance of the associated c-valued physical quantityÕ(n, ξ|ψ), i.e., Moreover, using again Eq. (14), the statistical deviation (it is mistakenly written as statistical distance in the version published in Phys. Rev. A 103, 022215 (2021) which has a completely different meaning) between two quantum observablesÂ andB over a quantum state |ψ can be expressed as the l 2 -norm of the associated c-valued physical quantities, i.e.: The statistical deviation between two observables plays a central role in the derivations of the error-disturbance and joint-measurement uncertainty relations [38,39,[49][50][51]. We shall show in Sec. IV B that Eq. (21) provides a mathematical link between our formalism and that of Johansen [37] and Hall [38,39] with regard to the interpretation of the decomposition of the c-valued physical quantities defined in Eq. (3) as describing an optimal scheme of estimation. Note further that, sinceÂ andB are in general noncommuting so that they cannot be jointly measured, the right-hand side of Eq. From Eqs. (13) and (10), we also obtain the following result. Consider two arbitrary operatorsÂ,B, and a function f (Ĉ) of a Hermitian operatorĈ. Taking the complete set of eigenstates {|c n } ofĈ as the reference coordinate basis to define the c-valued physical quantities, we obtain where in the last line we have usedC(c n , ξ|ψ) = c n . As a specific example, forÂ =B =p and C =q, we have ψ| 1 . Finally, assume further that ξ satisfies ξ 3 = 0. Then we obtain, for an arbitrary pair of operatorsÂ andB, See Appendix A for a proof.
All the above results show that the class of c-valued physical quantities defined in Eq.
(3) provides a real-deterministic hidden variable model for quantum expectation value of a certain class of operators. Note that Hosoya et al. in [19] and Hall et al. in [46] also obtained a representation of the product of two arbitrary operators in terms of the average of weak values. Indeed inserting Eq. (3) into Eq. (23) we regain their result: However, in their representation, the expectation value of the product of two operators are reconstructed directly from the average of the product of the complex weak values even when the operatorsÂ andB are Hermitian, i.e., the hidden variables are given by complex-valued variables. By contrast, within our model, as can be seen in Eqs. (13) or (23) (17) shows that there are two ways to achieve this, i.e., by directly averaging the c-valued kinetic en-ergyH(q, ξ|ψ) or averagingp(q, ξ|ψ) 2 /2m computed from the c-valued momentump(q, ξ|ψ).
Note however that while their averages are equal, as shown in Eq. (12) Finally, let us consider a mixed state or density matrix̺ which can be decomposed in terms of a set of pure states as̺ = µ Pr(ψ µ ) |ψ µ ψ µ |. It can be prepared e.g. by sampling the pure state |ψ µ with a probability Pr(ψ µ ) and deleting the record about the index µ. For a coordinate basis {|φ n }, the preparation thus samplesÕ(φ n , ξ|ψ µ ) with a joint probability Pr(ψ µ , φ n ) . = Pr(ψ µ )| φ n |ψ µ | 2 . Noting this, the results presented so far for pure states can then be generalized to the above statistical mixture of c-valued physical quantities by using the Bayes rule in conventional probability theory. As an example, computing the correlation between the mixture of two c-valued physical quantitiesÃ andB we obtain: where we have used Eq. (13).
A couple of notes are in order. First, the extension to mixture of pure states above does not employ the usual definition of the weak value for the pre-selected mixed state̺, which for an operatorÔ and a post-selected state |φ n is defined as [37,39,52] O w (φ n |̺) , we obtain the c-valued physical quantity associated with the mixed state that is independent of its pure states decomposition, i.e., where O w R (φ n |̺) and O w I (φ n |̺) are respectively the real and imaginary parts of weak value of O w (φ n |̺) with the pre-selected mixed state̺ defined in Eq. (26).
Here, we have used Eq. (4) to get the last equality, and ∆ 2Õ [ψ] and E 2 O [ψ] for an arbitrary c-valued physical quantityÕ(φ n , ξ|ψ) are defined as where in the third line we have again used Eq. (4). Hence, ∆ 2Õ [ψ] and E 2 O [ψ] are respectively the variance of the ξ-independent and ξ-dependent terms of the c-valued physical quantitỹ O(φ n , ξ|ψ), which are given by the variance of the real and imaginary parts of the weak value O w (φ n |ψ).

Upon inserting Eq. (28) into Eq. (20)
, we thus obtain the following general decomposition of quantum variance of a Hermitian operatorÔ over a quantum state |ψ into the variance of the ξ-independent term and the ξ-dependent error term of the associated c-valued physical quantityÕ(φ n , ξ|ψ): Notice that the left-hand side of Eq. (30) is basis-independent while the decomposition on the right-hand side is basis-dependent. Hence, for a given quantum variance σ hence, we have a spectral decompositionB = n b n |b n b n |. Let us take {|b n } as the reference coordinate basis providing the context to define the c-valued physical quantities.
We thereby obtain, from Eq. (3), Hence, using the reference basis {|b n }, noting Eqs. (28), (29) and (31), the variance ofÃ andB are decomposed as Next, since ∆ 2Õ [ψ] and E 2 O [ψ] are just the variances of c-valued -i.e., classical, commuting -variables, they must satisfy the usual uncertainty relation for real-valued classical random variables. To this end, recall that, from conventional probability theory, one has, using the Cauchy-Schwartz inequality, the following general relation between the variances of any two classical random variables X and Y, and their corresponding covariance: We show below that imposing the above classical uncertainty relation to the two c-valued physical quantitiesÃ(b n , ξ|ψ) andB(b n , ξ|ψ) in Eq. (31) leads to the Schrödinger and Kennard-Robertson lower bounds in the KRS uncertainty relation, separately.
First, taking X = A w R (b n |ψ) and Y = B w R (b n |ψ) = b n into Eq. (33), one has: where we have used a corollary of Eq. (6) that O w R (b n |ψ) = ψ|Ô|ψ . Next, putting X = (ξ/ )A w I (b n |ψ) and Y = B w R (b n |ψ) = b n into Eq. (33), one obtains, Note importantly that in the above derivations, we have used the fact that bothÂ andB are Hermitian. Hence, by taking the eigenstates {|b n } ofB as the reference coordinate basis, it becomes clear that the Schrödinger lower bound arises from the fluctuation of the ξ-independent term, and the Kennard-Robertson lower bound arises from the fluctuation of the ξ-dependent error term ofÃ(b n , ξ|ψ). We can instead of course take the complete set of eigenvectors {|a } ofÂ as the reference coordinate basis, so that instead of Eq. (31) we obtainÃ(a n , ξ|ψ) = a n andB(a n , ξ|ψ) = B w R (a n |ψ)+ ξ B w I (a n |ψ). In this way, we interchange the role ofÃ andB in the above derivations leading to the same Schrödinger and Kennard-Robertson lower bounds. We emphasize that while all the terms on left-hand sides of Eqs.  A different well-known method to map the q-valued quantum operators into c-valued variables is to employ the quasiprobability representations [54][55][56]. In fact, quasiprobability representation is deliberately devised to express the quantum expectation values of arbitrary operators evocative of classical average over c-valued variables. This approach therefore also leads to the expression of the KRS uncertainty relation in terms of c-valued variables. However, the quasiprobability distributions associated with quantum preparation and measurement may take on negative or even complex values, or is highly irregular. Moreover, from classical statistical point of view, it is not fully clear how the uncertainty relation, and in particular the Schrödinger and Kennard-Robertson lower bounds, arise within the quasiprobability representations. We note that within our formalism, from Eq. (3), for two arbitrary operatorsÂ andB, a pre-selected state |ψ , and a reference coordinate basis {|φ n }, we always have a well-defined non-negative joint probability distribution of the associated c-valued physical quantities Pr(Ã,B|φ n , ξ, ψ) We shall show below that whenÂ andB are noncommuting, the above joint probability cannot take an arbitrary form, which, in turn, underlies the uncertainty relation. First, assume that the reference coordinate basis is given by the complete set of eigenvec-tors {|b n } ofB with the associated set of eigenvalues {b n }. Then, one hasB(b n , ξ|ψ) = b n which is distributed as Pr(b n |ψ) = | b n |ψ | 2 . On the other hand, in this basis, the strength of the error term in the c-valued physical quantityÃ(b n , ξ|ψ) associated with a Hermitian op-eratorÂ, which is given by the imaginary part of the weak value A w I (b n |ψ), can be expressed as Here, we have used (d/dθ) |ψ ψ| .
|ψ ψ|] to arrive at the second equality, that is, we assumed that the Hermitian operatorÂ generates a unitary transformation to the state |ψ with the evolution parameter θ. Equation (38) shows that the error term of thus takes the form:Ã Eq. (40) reveals an intrinsic correlation or interdependence between the distribution of B(b n , ξ|ψ) = b n which is given by Pr(b n |ψ) = | b n |ψ | 2 , and the functional form ofÃ(b n , ξ|ψ).
This is a form of epistemic or statistical restriction, in the sense that the allowed distribution ofB(b n , ξ|ψ) = b n is restricted or parameterized by the functional form ofÃ(b n , ξ|ψ). This in turn will therefore restrict the allowed joint distribution of (Ã,B) defined in Eq. (37).
Note that when this epistemic restriction vanishes, namely when To get a better intuition on such a specific epistemic restriction, let us compare it with what happens in classical mechanics. For this purpose, take the complete set of eigenvectors {|q } of the position operatorq as the coordinate basis, so that we haveq(q, ξ|ψ) = q. Now, let us again write the wave function associated with the pre-selected state |ψ in polar form, i.e., ψ(q) = q|ψ = ρ(q)e iS(q)/ so that we have Pr(q|ψ) = |ψ(q)| 2 = ρ(q). The c-valued momentum in the position coordinate basis then reads Note that the unitary operatorÛ = e − i pqo applied to the pre-selected state |ψ generates a spatial shift q o to the wave function ψ(q), so that we have ∂ q ρ = −∂ qo ρ. Upon inserting into Eq. (41), we obtainp(q, ξ|ψ) = ∂ q S(q) + ξ 2 ∂q o ρ(q) ρ(q) , in accord with the general relation of Eq. (40). Noting this, the error term of the c-valued momentum, i.e., the second term on the right-hand side of Eq. (41), thus reflects the variation of the ρ(q) due to the spatial shift of the wave function induced by the unitary transformationÛ = e − i pqo generated byp.
One can then see from Eq. (41) that the allowed distribution of position ρ(q) is indeed inherently restricted or fundamentally parameterized by the form of the underlying momentum fieldp(q, ξ|ψ). Operationally, it means that each spatial point q cannot be sampled with an arbitrary weight ρ(q) independent of the momentum fieldp(q, ξ|ψ) [47]. In a sharp contrast to this, in classical mechanics, the distribution of position can always be chosen freely independent of the underlying momentum field, i.e., each spatial point q in the momentum field can be weighted with an arbitrary ρ(q). In this sense, classical mechanics thus enjoys an epistemic freedom which is violated by a specific epistemic restriction embodied in Eq. (41). This epistemic restriction in turn constrains the allowed form of the joint distribution of (q,p).
As an illustrative example of such a statistical constraint in phase space, assume that the c-valued momentum takes the formp(q, ξ|ψ) = ∂ q S = p o , where p o is a constant. Hence, the ξ-dependent term is vanishing and the c-valued momentum has a distribution with a sharp peak atp = p o . Then, to comply with Eq. (41), we must have ∂ q ρ/ρ = 0 so that ρ(q) must be spatially uniform. Moreover, one also has S(q) = p o q, up to an unimportant constant.
The above situation exactly captures the case when the wave function is a plane wave, i.e., ψ(q) ∼ e ipoq/ . Hence, when the distribution ofp is sharp, ρ(q) = ρ(q) must be spatially uniform in accordance to the spirit of the Heisenberg uncertainty principle. Indeed, from the specific interdependence between the momentum field and the distribution of position expressed in Eq. (41), one can directly derive the KRS uncertainty relation for position and momentum [47,57,58]: where we have made use of Eq. (4).
B. Optimal estimation under epistemic restriction, Kennard-Robertson uncertainty relation, and classical limit With the above observation in mind, below we shall argue that the two terms decomposing the c-valued physical quantity in Eq. (3) can be interpreted as an epistemic decomposition describing an optimal (i.e., best) estimation under the epistemic restriction. Let us take again the complete set of eigenvectors {|b n } ofB as the reference coordinate basis so that we haveB(b n , ξ|ψ) = b n . Now, consider an agent who wishes to estimateÃ(b n , ξ|ψ) associated with a Hermitian operatorÂ and a pre-selected state |ψ , based on the information about b n sampled from the distribution Pr(b n |ψ) = | b n |ψ | 2 , which, as argued in Sec. IV A, is fundamentally parameterized byÃ(b n , ξ|ψ). Assume that the agent's estimate is given by i.e., the first term on the right-hand side of Eq. (3). The single-shot estimation error is thus given by the second term on the right-hand side of Eq. (3), i.e., For any ξ, the average of the error over the distribution of b n is vanishing: n ψ|[Π bn ,Â]|ψ = 0. In this sense, the above estimation scheme is unbiased.
One can then see from Eq. (43) that the estimation error is larger when the incompatibility betweenÂ and the reference coordinate basis {|b n } over the state |ψ is stronger.
Computing the mean squared (MS) estimation error one gets n dξǫÃ(b n , ξ|ψ) 2 χ(ξ)| b n |ψ | 2 where we have made use of Eq. (29) in the last equality. Within the above interpretation, the decomposition of quantum variance in Eq. (30) can therefore be read as the decomposition of the variance of the c-valued physical quantity into the precision (i.e., the spread) of the estimate given by ∆ 2Ã [ψ] and the accuracy of estimation given by the associated MS . Let us show that the above estimation scheme is optimal, i.e., it minimizes the MS estimation error. First, given a pre-selected state |ψ and a reference coordinate basis {|b n }, let TÃ(b n |ψ) be an estimator forÃ(b n , ξ|ψ) based on information on b n . The associated MS estimation error can be directly computed to give, noting Eqs. (3) and (4), One can see that, as claimed, the minimum is obtained when the estimator is given by so that the first term on the right-hand side of Eq. (45) vanishes. Moreover, the minimum MS estimation error is indeed given by Eq. (44).
This interpretation of the real and imaginary parts of the weak value which constitute the c-valued physical quantity defined in Eq. (3) is similar to that proposed by Johansen [37] and Hall [38,39]. See also Refs. [45,53,59].
where the last equality is just Eq. (29). Multiplying E 2 A [ψ] to both sides of Eq. (47) and noting Eq. (35), we thus obtain In this sense, by interpretingÃ(b n , ξ|ψ) and B o [ψ] as the parameters for the distribution Pr(b n |ψ) = | b n |ψ | 2 , the Kennard-Robertson lower bound can be seen as a fundamental limitation of the simultaneous estimation of the two parameters based on information oñ B(b n , ξ|ψ) = b n . In particular, it describes the trade-off between the associated MS estimation errors.
Finally, within the above epistemic interpretation of the decomposition of the c-valued physical quantity in terms of the estimation under epistemic restriction, the limit of vanishing estimation error is obtained when ξ → 0. In this limit, information on the coordinate basis {|φ n } is sufficient to determine the c-valued physical quantities associated with arbitrary operators as in classical mechanics. Of particular interest is the estimation of the c-valued momentump(q, ξ|ψ) based on information on the conjugate positionq(q, ξ|ψ) = q. In this case, the c-valued momentum is decomposed as in Eq. (41), wherein the first term on the right-hand side is the optimal estimate and the second term is the single-shot estimation error. In the limit ξ → 0, which is equivalent to → 0 as per Eq. (4), the estimation error is vanishing so that we obtain lim ξ→0p (q, ξ|ψ) = ∂ q S(q).
Hence, in this physical regime, the epistemic restriction disappears, namely, the momen- A similar speculation concerning the weak value is suggested in Ref. [19] and is applied to discuss the Hardy paradox. Note however that, as argued in Sec. IV A, even for such a realistic interpretation ofÕ(φ n , ξ|ψ), its decomposition into two parts, i.e., into the ξ-independent and ξ-dependent terms in Eq. (3), is epistemic (informational). Namely, the ξ-independent term should be interpreted as the optimal estimate of the c-valued physical quantity, while the ξ-dependent term is the associated single-shot estimation error. Hence, the specific decomposition in Eq. (3) is artificially made in the agent's mind to optimally organize her experience. Within this epistemic interpretation, as shown in Eq. (48), the Kennard-Robertson uncertainty relation describes the trade-off between the MS errors of the joint estimation of two parameters associated with two noncommuting observables.
Since the definition of c-valued quantities depends on the context determined by the choice of the reference coordinate basis for post-selection measurement, it is not sufficient for a measurement independent reformulation of quantum mechanics. However, c-valued quantities may be defined as counterfactual entities independent of measurement by choosing a physically meaningful reference basis. It may then be possible to reconstruct the whole of quantum mechanics, not just the quantum uncertainty relations, from scratch, starting from the c-valued physical quantities and imposing a set of plausible physical principles. For infinite dimensional continuous variable quantum systems, this program was shown to be viable in Ref. [47]. In this reconstruction, we have assumed the c-valued momentump(q, ξ|ψ) given in Eq. (41) as the counterfactual momentum field associated with a preparation. It is interesting to ask if this reconstruction program can be generalized to discrete variable quantum systems with finite dimension of Hilbert space by employing the c-valued physical quantity defined in Eq. (3), and if the specific forms of the two decomposing terms in Eq.
(3) can be justified by the principle of estimation independence as for the two terms in Eq.
(41) [57,58]. Such a reconstruction will lend further support to the epistemic interpretation of quantum mechanics advanced in Ref. [35]. Moreover, it is intriguing to further study if the counterfactual c-valued physical quantities can shed some fresh light in the discussion of the Hardy paradox [60] relative to that based directly on the weak value [19,[22][23][24][25].
Several other questions are worth pursuing in the future. First, by introducing a supplementary global random variable ξ, we have been able to transform the complementarity between two incompatible quantum observables into the complementarity between the error terms of the associated two c-valued physical quantities. It is thus tempting to exploit this intuition to study the wave-particle duality and quantum coherence. It is also intriguing to explore the possible empirical implications of such a global random variable. Especially, assuming that ξ characterizes the randomness of each single repetition of the weak measurement, we want to see ifÕ(φ n , ξ|ψ) is realizable in a single-shot weak measurement with post-selection and to study its distribution. We also would like to make a detailed comparison between the representation based on c-valued physical quantities with the approach based on quasiprobability, and to elaborate its computational possibility [61]. In particular it is interesting the compare the present formalism with the complex-valued Kirkwood-Dirac quasiprobability [62][63][64], or its real part, i.e., the Terletsky-Margenou-Hill quasiprobability [65,66], which are also deeply related to the concept of weak value [14,15,59,67,68].
Finally, we want to further investigate the interpretation of the c-valued physical quantity in terms of estimation under epistemic restriction and its possible application in quantum parameter estimation [69,70].

Acknowledgments
Appendix A: Derivation of Eq. (23) Let us assume additionally that the third moment of ξ is vanishing, i.e., ξ 3 = 0. For example, assume that ξ is a binary random variable ξ = ± with equal probability. We then obtain the following result concerning the representation of the commutator between two operators in terms of the statistics of the associated real-deterministic c-valued physical quantities: ξ Ã (φ n , −ξ|ψ)B(φ n , ξ|ψ) where to obtain the second equality we have used the mathematical identity Im{a * b} = Re{a}Im{b} − Im{a}Re{b} for two complex numbers a and b. As a corollary, if bothÂ and as claimed in Eq. (23). We note that the decomposition on the right-hand side is basisdependent, i.e., contextual with respect to the choice of post-selection measurement {|φ n } of the associated weak value measurement, while the left-hand side is not. One can get a simplified expression by taking the complete eigenstates {|a n } ofÂ as the reference