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Operational characterization of general quantum correlation via complex weak value measurement

The last two decades have witnessed significant progress on the understanding the quantum correlation more general than entanglement, wherein even a separable state may yield correlation that cannot be emulated by any classical object. Such a general nonclassical correlation is not only intriguing from the fundamental point of view, but it has also been recognized as a resource in a variety of quantum information processing tasks and quantum technology. Here, we propose a characterization of the general quantum correlation in bipartite system in terms of direct laboratory operations using weak measurement with postselection. We define a quantity based on the imaginary part of weak values obtained via weak measurement of a local basis followed by a postselection of another local basis, and an optimization procedure over all possible choices of the two bases. We show that it satisfies certain desirable requirements for a quantifier of general quantum correlations. It may be statistically interpreted as the minimum genuine quantum share of uncertainty. It is a faithful witness of entanglement for general pure states, giving an observable lower bound to a scaled square root of the linear entropy of entanglement. We then suggest an information theoretic interpretation of the general quantum correlation in a multipartite state as the minimum mean absolute error in an optimal estimation of any local measurement basis, based on the outcomes of local projective measurement on the state, in the worst case scenario.


I. INTRODUCTION
Nonclassical correlation in multipartite quantum systems was originally associated with entangled states, namely states that cannot be prepared by any set of local operation and classical communication (LOCC) [1]. Studies in the last couple of decades however show that most unentangled (i.e., separable) states are somewhat nonclassically correlated, manifested in the various forms of discord-like quantum correlations . See Refs. [27][28][29] for recent reviews. Conceptually, these general quantum correlations in which entanglement is a subset, arise from the noncommutativity between the multipartite quantum state and some set of local quantum observables, rather than from the nonseparability of the state. The nonclassical correlations beyond entanglement have received a growing attention both from the fundamental view point and also practically to better understand the physical origin of the quantum advantages in certain information processing tasks and schemes of quantum technology which consume very little or no entanglement [30][31][32][33][34][35][36][37]. Different approaches have been proposed to characterize and quantify the general quantum correlation, using quantum information theoretical concepts which take on an input of the state of the multipartite system. They thus indirectly need a full quantum state tomography for its evaluation.
It remains intriguing whether it is possible to characterize and quantify general quantum correlation directly in terms of transparent laboratory operations.
On the other hand, there is a measurement scheme in which the noncommutativity between the state and the observable appears directly at the pointer of the measuring device.
Consider the weak measurement of a Hermitian observable O with the preselected state ̺ followed with the postselection on the state |φ via a normal, i.e., strong projective measurement. One then obtains the following complex weak value [38][39][40]: with its real and imaginary parts that can be directly read-off respectively from the average shift of the position and momentum of the pointer of the measuring device [41,42]. The complex nature of the weak value has been exploited recently to devise a method for direct measurement of a quantum state or its quasiprobability representation [43][44][45]. Notice in particular that the imaginary part of the weak value, i.e., Im{O w (φ|̺)} = 1 2i directly captures the noncommutativity between the state and the observable which is the origin of various nonclassical properties of quantum systems encoded in the quantum state.
It is thus natural to ask if one can use the operational scheme of weak measurement with postselection to directly characterize and quantify the quantum correlation beyond entanglement arising from the noncommutativity between the multipartite state and some set of local observables.
In this paper, we argue for an affirmative answer to the above question. First, given a preselected multipartite state, we operationally define a quantity from the imaginary part of the weak values obtained by making weak measurement over a complete set of local orthonormal basis of the subsystems, followed by postselection over the product of the local bases, and optimized over all possible choices of these two bases. We demonstrate that it satisfies certain plausible properties expected for a quantifier of quantum correlation beyond entanglement. It sets a lower bound to the negativity of quantumness [16,46]. We argue that it may be interpreted as the minimum genuine quantum share out of the total uncertainty of the local measurement basis in the multipartite state. For general pure states, it gives an observable lower bound, hence a witness, to the scaled square root of the linear entropy of entanglement. Our approach thus offers a definition of general quantum correlation in terms of direct laboratory operations based on weak measurement with postselection, which might suggest a fresh insight into its role as a resource in quantum information processing.
Moreover, it leads to the following information theoretical interpretation of the general quantum correlation: a multipartite state is quantumly correlated if and only if the optimal estimation of any local basis, based on the outcomes of the local projective measurement over the state, must yield a nonvanishing mean absolute error in the worst case scenario, with the minimum value that is given by the quantum correlation. On the other hand, a multipartite state is classical devoid of quantum correlation if and only if, local measurement over the state can be used to make a sharp optimal estimation of a certain local basis.
is the density matrix of the subsystem A(B) on H A(B) , and {p k } are mixing probabilities: p k ≥ 0, k p k = 1. Any other states, i.e., nonseparable states, are entangled. For pure states, separability is equivalent to the absence of nonclassical correlation. Remarkably, allowing impurity to the bipartite state may lead to nonclassical correlations even when it is separable. That is, various different schemes have been revealed in the last couple of decades which show that mixed separable states may contain correlation that cannot be accessed by any classical object . For instance, such a separable state may yield a global modification as a response to any local measurement [2,3,15].
By contrast, classical intuition suggests that for a genuine classical multipartite state, there is at least a local measurement which does not lead to a global modification to the state. In the bipartite setting, there are two classes of states which conform to the above intuition [9,33]. One is a class of so-called classical-quantum (or, A−classically correlated) states, which are those that can be expressed as where {|k A } is an orthonormal basis of the Hilbert space H A of subsystem A, {̺ k B } ∈ H B are states of subsystem B, and {p k } are probabilities, i.e., p k ≥ 0, k p k = 1. The quantumclassical (or, B-classically correlated) state is defined analogously by exchanging the role of the parties A and B. The other class consists of bipartite states with density matrix which take the following form: where {|k A } and {|l B } are respectively orthonormal basis of subsystem A and B, and {p kl } are joint probabilities so that p kl ≥ 0, k,l p kl = 1. Such states are called classical-classical (or, totally classically correlated) states. One can see that applying local projective (von = |x x| is one-dimensional projector onto the subspace spanned by |x ), respectively, to the classical-quantum states and classical-classical states, without learning or recording the outcomes, leave them unmodified, This can be seen as due to the fact that the local measurement basis and the bipartite state commute, i.e., , ̺ CC AB ] = 0, for all k, l. Any bipartite state that cannot be represented either in the forms of Eqs. (3) or (4) thus contain nonclassical correlation arising from the noncommutativity between the state and the local measurement basis, even if it is separable.
Various mathematical characterizations of the classical bipartite states of the types of Eqs.
(3) and (4) have led researchers to develop different quantifiers and measures of quantum correlation beyond entanglement, by applying quantum information theoretical concepts: quantum mutual information, von-Neumann entropy, distance and infidelity between two density matrices, Wigner-Yanase skew information or quantum Fisher information [28]. All these quantifiers essentially quantify the difference between the bipartite state under scrutiny and the classical bipartite states of Eqs. (3) and (4). Moreover, they indirectly or directly reflects the noncommutativity between the bipartite states and some set of local observables.
To evaluate the nonclassical correlation in a bipartite quantum state using these quantifiers, we typically need an input of the associated density matrix, thus requires a tomography of the full quantum state. In order to better understand the operational meaning of the quantum correlation beyond entanglement, it is desirable to have a quantifier of general quantum correlation whose definition directly and transparently translates into laboratory operations.
On the other hand, as noted in the previous section, the imaginary part of the weak value can be used to directly probe the noncommutativity between a preselected bipartite quantum state and the local measurement basis as follows. Suppose we wish to probe the noncommutativity between a bipartite state ̺ AB and a local measurement basis {Π k A } on subsystem A. To do this, one first makes a weak measurement of Π k A ⊗I B with the preselected state ̺ AB followed by a postselection on |φ AB , reads the imaginary part of the weak value from the average shift of the pointer momentum, and multiply it with the probability of the successful postselection, to get We note that the noncommutativity between the state and a Hermitian observable can also be directly checked via quantum variance capturing the uncertainty of the observable in the state. However, quantum variance also takes the contribution of the uncertainty due to the statistical, i.e., classical mixing of states, so that it is not genuinely quantum.

VALUE MEASUREMENT
We want to devise a quantifier of general quantum correlation in a bipartite state from |φ AB ] defined in Eq. (5) so that it can be observed directly using weak measurement with postselection. To this end, given an arbitrary bipartite quantum state ̺ AB , let us define the following quantity which maps the state to a nonnegative real number: ] is intended to quantify the nonclassical correlation in the bipartite state ̺ AB arising from the noncommutativity between ̺ AB and any local measurement basis {Π a A } of the subsystem A. Similarly, to quantify the nonclassical correlation in the bipartite state ̺ AB which arises from the noncommutativity between ̺ AB and any local we define the following quantity which maps the density matrix to a nonnegative real number: where, in this case, the minimization is carried out over all possible local measurement bases For a similar reason, we emphasize that the search for the minimum over the local measurement bases is performed after the maximization over the postselection bases is done. Otherwise, for each postselec- , maximizing over all possible postselection bases {|φ A , φ B } means that we are seeking for the largest incompatibility between On the other hand, the minimization over the local measurement bases ) means that we search for those classical-quantum state of Eq. (3) (respectively, classical-classical state of Eq. (4)) that is We further emphasize that the projective postselection measurement is over product of  (7), respectively, as one-sided and two-sided w-nonclassical correlation. Finally, the above definition can be extended to more than two parties straightforwardly, while retaining clear operational meaning. For example, the one-sided w-nonclassical where the postselection on the subsystem C is given by the orthornormal basis of subsystem C. Let us mention that completely different approaches to characterize general quantum correlation based on weak measurement but without postselection, and thus independent of the concept of weak value, were reported in Refs. [47,48].
Consider first the one-sided w-nonclassical correlation C Q A [̺ AB ] defined in Eq. (6). We argue that it satisfies the following desirable properties for a quantifier of quantum correlation [28]: is any unitary operator applying locally on subsystem A(B).
(iii) Nonincreasing under any local completely positive trace preserving (CPTP) operation Λ B on subsystem B, namely on the party whose nonclassical correlation is not being Let us sketch the proofs of the above properties and discuss a couple of others.
To establish property (i), first, let us suppose that the bipartite state ̺ AB belongs to the class of classical-quantum state, namely there exists an orthonormal basis {Π k A = |k k| A } on the subsystem A so that ̺ AB can be expressed as in Eq. (3). Then, we can choose The property (ii) of invariant under local unitary transformation can be shown as follows: Here, we have inserted an identity ( To get the third line, we have defined a new postselection basis: Noting that the above transformation of bases do not change the sets B {|φ A ⊗|φ B } and B {Π a A } over which we respectively perform the maximization and the minimization in Eq. (6), we thus have . This observation gives the last equality in Eq. (8).
To establish the property (iii), first, we recall that according to the Stinespring's theorem, any CPTP operation or quantum channel can be implemented by a dilation on a larger Hilbert space, wherein the system is made contact with an ancilla in a state ̺ E on the Hilbert space H E , let them interact via some global unitary, and then followed by partial tracing over the ancilla as: U BE is the unitary interaction applying on the subsystem B and the ancilla E. Using this, we then have where the basis {|φ E } of the Hilbert space H E of the ancilla inserted in the second line is arbitrary, and we have chosen {|φ E } so that the unitarily transformed postselection basis of the whole system-ancilla is again factorizable, i.e., as claimed. Note that |φ ′ B now depends on the index φ E . Finally, from Eq. (9), noting that φ E |̺ E |φ E is nonnegative and real, and φ E φ E |̺ E |φ E = 1, we obtain: where, in the last line, we have made use of the fact that, for each φ E , the set which is independent of φ E . Besides satisfying the above three plausible constraints, the one-sided w-nonclassical cor- where {p k } are probabilities, due to triangle inequality and the fact that p k ≥ 0. Moreover, the one-sided w-nonclassical correlation is nonincreasing when one or more of the parties are discarded (ignored). To show this, without losing generality, consider for example a tripartite state ̺ ABC . One then has where The equality is reached when the subsystem C is totally uncorrelated with the rest, i.e., , by virtue of the fact that φ C |̺ C |φ C is real and nonnegative and φ C φ C |̺ C |φ C = 1. Hence, adding or removing an (independent) ancilla does not change the w-nonclassical correlation as intuitively expected. The above property straightforwardly generalizes to any number of parties where Λ C is a completely positive trace preserving operation on C. One can also easily see that C Q AB [̺ AB ] is convex. Moreover, discarding one or more parties does not increase the wnonclassical correlation, e.g., we have Beside these, the two-sided w-nonclassical correlation satisfies the following inequality: by noting that c, It is the state of AB after the strong projective measurement of the local basis {Π c C } on ̺ ABC without reading the outcomes.
Finally, another important requirement for a quantifier to be a bonafide measure of quantum correlation is that for pure states, it should reduce to a corresponding measure of entanglement [28]. This captures the intuition that for pure states, separablity is equivalent to no nonclassical correlation. We have not yet been able to clarify this important issue.
However, in the next section, we prove that for general pure states of arbitrary finite dimension, the one-sided w-nonclassical correlation defined in Eq. (6) gives an observable lower bound to a scaled linear entropy of entanglement, which for two qubits pure state reduces to the entanglement concurrence [49]. Hence, for pure states, the w-coherence can be seen as a faithful witness of entanglement.
Let us further show that, for any generic mixed state, the two-sided w-nonclassical correlation sets an observable lower bound to the negativity of quantumness, a measure of general quantum correlation which quantifies the amount of entanglement that can be activated via a local measurement of the subsystem [16,46]. First, starting from the definition of the two-sided w-nonclassical correlation in Eq. (7), we have where we have inserted the completeness relations a One the other hand, using Cauchy-Schwartz inequality, we have where we have again used the completeness relation we finally obtain The quantity on the right-hand side is just the measure of general quantum correlation based on l 1 -norm measure of (local) coherence [28] which is equal to twice of the total (twosided) negativity of quantumness [16,46]. We emphasize that unlike the measure of general quantum correlation based on l 1 -norm (local) coherence and negativity of quantumness, the w-nonclassical correlation is in principle directly observable via weak measurement and postselection.
We proceed to give a concrete computation of the w-nonclassical correlation in a simple bipartite state. We first note that, as is also the case for the quantifiers of general quantum correlation proposed in the literature, the calculation of the w-nonclassical correlation defined in Eqs. (6) and (7) is in general analytically intractable, involving optimization over all possible relevant bases, which in general takes the form of an optimization of multivariable nonlinear function. For example, consider the following maximally entangled two-qubit state . Inserting all the above ingredients into the one-sided nonclassical See Appendix A for the detailed derivation of the above trigonometric function that has to be optimized, i.e., maximized over the parameters (α A , α B , β A , β B ) and then followed by choosing the minimum over (θ A , η A ).
Next, as an example of a separable state with a nonvanishing one-sided w-nonclassical correlation, we consider the Werner state for 2 × 2 dimension: It is known that the above Werner state is separable for p < 1/3. The one-sided wnonclassical correlation can be straightforwardly computed to get, noting Eq. (19), where the first equality is obtained directly from the definition in Eq. (6). Hence, it is nonvanishing even in the regime when the Werner state is separable. We note that the above value of the w-nonclassical correlation for the Werner state is equal to the square root of the geometric discord based on Hilbert-Schmidt distance [34].

IV. STATISTICAL AND INFORMATION THEORETIC MEANING
A. W-nonclassical correlation as the witness for genuine local quantum uncertainty and pure state entanglement Recall that the one-sided and two-sided w-nonclassical correlations are defined via the noncommutativity between the local measurement bases and the multipartite state. Such noncommutativity is one of the sources of uncertainty in quantum measurement. It is therefore instructive to discuss the relation between the w-nonclassical correlations and the local quantum uncertainty. Let us first consider the one-sided w-nonclassical correlation.
Then, as shown in the Appendix B, we obtain the following inequality where ∆ 2Ô [̺] . = Tr{Ô 2 ̺} − (Tr{Ô̺}) 2 is the quantum variance ofÔ in the state ̺ which takes into account both the genuine quantum uncertainty arising from the noncommutativity between O and ̺, and the classical uncertainty associated with classical mixing in the state ̺. Similarly, for the two-sided w-nonclassical correlation, we obtain and similarly, for any local measurement bases The w-nonclassical correlation can therefore be seen as one of the sources of the quantum uncertainty in the measurement of the local basis which arises from the noncommutativity between the local basis and the multipartite state.
The above observation suggests to interpret C Q A [̺ AB ] and C Q AB [̺ AB ] as the minimum genuine quantum share of the local uncertainty, out of the total local quantum uncertainty quantified by the quantum standard deviation. Recall that the latter also includes the uncertainty arising from the classical mixing of states, and possibly from other sources. To this end, it is intriguing to compare the w-nonclassical correlation with the Wigner-Yanase skew information [50], which is also based on the noncommutativity between the state and a Hermitian observable and has been argued to quantify the genuine quantum share of uncertainty arising from quantum noncommutativity [51]. Let us mention that the minimum value of the Wigner-Yanase skew information over certain set of local nondegenerate observables is proposed in Ref. [25] as a quantifier of nonclassical correlation beyond entanglement. We note that unlike the Wigner-Yanase skew information, w-nonclassical correlation C Q A [̺ AB ] and C Q AB [̺ AB ] are operationally appealing, in that they have transparent interpretation in terms of direct laboratory operations based on weak measurement and postselection.
Next, noting that Tr{(Π a A ⊗ I B ) 2 ̺ AB } = Tr{(Π a A ⊗ I B )̺ AB } = p(a|̺ A ; Π a A ) is just the probability to get the outcome a in the local projective measurement {Π a A } over the state ̺ A = Tr B {̺ AB } of the subsystem A, and inserting into Eq. (22), we obtain where we have again used the Cauchy-Schwartz inequality and the normalization a p(a|̺ A ; Π a A ) = 1 to get the second line, and is the Tsallis entropy with the entropy index 2.
Let us further show that the minimum of the Tsallis entropy on the right-hand side of Eq.
(24) is obtained when the local measurement basis {Π a A } is just given by the eigenprojectors of the reduced density matrix ̺ A . Suppose that the reduced density matrix has the following spectral decomposition: Inserting this into the definition of the Tsallis entropy in Eq. (25), and noting the convexity of the quadratic function, we get, using the Jensen inequality, where we have used the normalization a Tr{Π Finally, notice that the Tsallis entropy S 2 [{π j (̺ A )}] on the right-hand side of Eq. (27) is just the linear entropy associated with the reduced density matrix ̺ A = j π j (̺ A )Π η j A , i.e., It quantifies the mixedness or impurity of the reduced density matrix ̺ A = Tr B ̺ AB . For a pure bipartite state |ψ AB , the linear entropy can be used as a measure of entanglement, i.e., Eqs. (27) and (28) therefore show that for general pure bipartite states |ψ AB , the one- can be seen as a witness for the quantum entanglement in |ψ AB , namely, it gives a lower bound to the scaled square root of the linear entropy of entanglement as Moreover, since C Q A [|ψ AB ψ AB |] is vanishing if and only if |ψ AB is factorizable, it is a faithful witness for the pure state entanglement. We note that for two-qubit pure state, denoting the Schmidt coefficients as λ + and λ − , the right-hand side of Eq. (29) is just given by the entanglement concurrence [49], i.e., d  i.e., We then consider the worst case scenario wherein the total sum of mean absolute error of the Finally, among these worst case scenarios, we choose the local basis {Π a A } so that the total mean absolute estimation error is the least (i.e., the best of the worst scenario). This gives us the one-sided w-nonclassical correlation of Eq. (6), i.e., This is just a minimax estimation problem. An analogous interpretation can be made for the two-sided w-nonclassical correlation C Q AB [̺ AB ] of Eq. (7). The above information theoretic interpretation in terms of optimal estimation of local basis based on local measurement on the multipartite state, suggests a new characterization of the quantum versus classically correlated multipartite states of Eqs. (3) or (4) as follows.
A multipartite state is quantumly correlated if and only if the optimal estimation of any local basis based on the local projective measurement on the state must yield a nonvanishing mean absolute error in the worst case scenario, with the minimum value that is given by the w-nonclassical correlation. By contrast, a multipartite state is classical devoid of quantum correlation, if and only if the outcomes of local projective measurement on it can be used to make a sharp (i.e., with vanishing estimation error) optimal estimation of a certain local basis. Such a negative characterization of general quantum correlation is similar to the characterization of quantum correlation expressed by no-go theorem for quantum local broadcasting [33].

V. SUMMARY AND REMARKS
We have proposed a characterization of nonclassical correlation beyond entanglement in terms of transparent and direct laboratory operations. We operationally defined a quantity based on the imaginary part of weak values obtained via the weak measurement and postselection, together with an optimization procedure, which captures the minimum incompatibility between the multipartite state under scrutiny and any local measurement basis.
We showed that it satisfies certain desirable requirements for a quantifier of general nonclassical correlation in the multipartite state. We argued that it may be statistically interpreted as the minimum genuine quantum part of local uncertainty arising from the incompatibility between the local measurement basis and the multipartite state. It is a faithful witness for pure state entanglement giving an observable lower bound to a scaled square root of the linear entropy of entanglement. Moreover, we have offered an information theoretic interpretation of the nonclassical correlation as the minimum mean absolute error of the optimal estimation of a local basis, based on the outcomes of local projective measurement on the multipartite state, in the worst case scenario.
Our results suggest intriguing fundamental links between the quantum-classical division and contrast captured by general nonclassical correlation, and the concept of weak value in the multipartite setting. Note that the ability of the scheme of weak measurement with postselection to directly characterize general nonclassical correlation is related to the fact that a similar scheme can be used to directly observe the quantum state or its equivalent quasiprobability representation [41,42], in which, the nonclassical features of quantum systems, including general nonclassical correlation, are encoded. In both schemes for the direct measurement of quantum states and direct characterization of general quantum correlation, the complex nature of the weak value, especially the imaginary part which captures the noncommutativity between the quantum state and observable, are crucial. It is hoped that the weak measurement with postselection may offer a fresh operational framework to characterize and quantify other nonclassical features of quantum mechanics.

Acknowledgments
Appendix A: Computation of Eq. (19) Let us first compute each term in the sum in Eq. (19) for different elements of the local measurement and postselection bases. From Eq. (18), it is convenient to express the local measurement basis as One then obtains, for the weak measurement of |µ+ µ+| A with the postselection state |φ A +, φ B + , using the expression for |µ+ µ+| A of Eq. (A1) and noting the expression for the postselection states in Eq. (17), = |Im{ φ A +, φ B + |(|µ+ µ+| A ⊗ I B )|Ψ + AB Ψ + AB |φ A +, φ B + }| = |I 00 ++ cos 2 θ A 2 + I 01 ++ sin θ A + I 10 ++ sin θ A + I 11 where each term on the right-hand side takes the form: On the other hand, for the weak measurement of |µ− µ−| A with the postselection state |φ A +, φ B + , using the expression for |µ− µ−| A of Eq. (A1), we get Similarly, one can compute the cases for the weak measurement of |µ A ± µ A ±| with the remaining postselection states |φ A +, φ B − , |φ A −, φ B + , and |φ A −, φ B − . One can further show by inspection that where, upon inserting Eq. (A3) into Eq. (A2), we have Inserting these terms into the definition of the one-sided w-correlation of Eq. (6), we obtain Eq. (19). Hence, we need to perform the maximization of nonlinear function of Eq. (19) over (α A , α B , β A , β B ), and followed by choosing the minimum over (θ A , η A ). We do this by using the subroutine fminsearch in a simple matlab code to get the maximum over (α A , α B , β A , β B ), followed with the application of subroutine min of matlab to search for the minimum over (θ A , η A ).
Appendix B: Proof of Eq. (22) First, we have, from Eq. (6), Here {|φ ⋆ } = {|φ ⋆ A , φ ⋆ B } is the postselection basis which achieves the maximum, and {Π a ⋆ A } is the local measurement basis which achieves the minimum. Moreover, we have applied the Cauchy-Schwartz inequality and the completeness relation for the postselection basis, i.e., φ ⋆ Tr{Π φ ⋆ AB ̺ AB } = 1, to get the third and fourth lines. Next, applying again the Cauchy-Schwartz uncertainty relation for bounded operators, |Tr{XY † }| 2 ≤ Tr{XX † }Tr{Y Y † }, to the numerator in the first term on the right-hand side of Eq. (B1), i.e., and using the completeness relation φ ⋆ Π φ ⋆ AB = I, we finally obtain Eq. (22), i.e., By following similar steps as above, we also obtain, for the two-sided w-nonclassical correlation, where {Π a ⋆ A ⊗ Π b ⋆ B } is the product of the local measurement bases for subsystems A and B which achieves the minimum.