Quantum correlation cost of the weak measurement
aa r X i v : . [ qu a n t - ph ] S e p Quantum correlation cost of the weak measurement
Jun Zhang, Shao-xiong Wu, Chang-shui Yu ∗ School of Physics and Optoelectronic Technology, Dalian University of Technology,Dalian 116024, China
Abstract
Quantum correlation cost (QCC) characterizing how much quantum correla-tion is used in a weak-measurement process is presented based on the tracenorm. It is shown that the QCC is related to the trace-norm-based quan-tum discord (TQD) by only a factor that is determined by the strength ofthe weak measurement, so it only catches partial quantumness of a quantumsystem compared with the TQD. We also find that the residual quantumnesscan be ‘extracted’ not only by the further von Neumann measurement, butalso by a sequence of infinitesimal weak measurements. As an example, wedemonstrate our outcomes by the Bell-diagonal state.
Keywords: weak measurement, trace norm, quantum correlation cost
1. Introduction
As one of the important quantum correlations, quantum entanglementhas been identified as an important physical resource in quantum informa-tion processing tasks (QIPTs) [1]. But in some QIPTs such as the robustquantum algorithm against the decoherence [2], the deterministic quantumcomputation with one quantum bit (DQC1) [3] etc, is there not any quantumentanglement, but quantum discord that was introduced in Ref. [4] and Ref.[5], respectively, has been shown to be able to grasp more quantumness thanentanglement. In recent years, the research on quantum discord has beenmade great progress in various fields [6–23].Quantum discord was originally defined by the difference between the to-tal correlation and the classical correlation that is obtained by the optimal ∗ Corresponding author. Tel: +86 41184706201
Email address: [email protected] (Chang-shui Yu)
Preprint submitted to Elsevier October 3, 2018 ocal measurements, since classical information is locally accessible, and canbe obtained without perturbing the state of the system [4]. Later it wasgeneralized to various cases by using different distance measures between thestate taken into account and the post-measured state [24–38]. In usual, thesementioned measurements are referred as to the von Neumann measurementwhich can cause collapse of the wave function and hence has strong influ-ences on (destroys) the initial state. In 1988, Aharonov et al. introduced theweak measurement [39] which was applied in many areas [40–43]. It was latergeneralized by Oreshkov and Brun [44] to the case with only preselection interms of measurement operator formalism and one of the important charac-teristics is that the strength of the measurement process can be controlledto be very weak, so the quantum state can be influenced weakly. Consid-ering such weak measurements in the quantification of quantum correlation,Singh and Pati proposed the super quantum discord [45]. It has been shownthat super discord is not less than the quantum discord. Recently, the superquantum discord has attracted increasing interests [46–50]. In particular, itis surprising that the weak measurement can extract extra quantumness andthe lost quantumness could even be resurrected by weak measurement [50].In this paper, we find very different phenomena from those in Refs.[45, 50]. Motivated by the super quantum discord, we present the quan-tum correlation cost (QCC) by employing the trace norm as the distancemeasure between the state of interests and post-weak-measured state. It isinteresting that the difference between the QCC and the quantum discordbased on trace norm (TQD) is in a factor that is determined by the strengthof the employed weak measurement. Instead of the extra quantumness [45],this factorization relation shows that the QCC quantifies how much quan-tumness of a quantum system is ‘extracted’ (used) by the employed weakmeasurements. In particular, we also find that the residual quantumnesswhich the weak measurement fails to extract will be extracted further, if thepost-weak-measured state is succeeded by the von Neumann measurementor by a sequence of infinitesimal weak measurements. In addition, all theseconclusions can also be generalized to (2 ⊗ d ) − dimensional systems and thecase of multiple-outcome weak measurements. As a demonstration, we studythe QCC of the Bell-diagonal state, which shows the consistency with ourconclusions. 2 . Quantum correlation cost and Residual quantumness Quantum correlation cost.-
To begin with, we would like to first introducethe original super quantum discord Q w ( ρ AB ) which is defined as Q w ( ρ AB ) = min Π Ai S w ( B |{ P A ( x ) } ) + S ( ρ A ) − S ( ρ AB ) , (1)where S ( · ) represents the von Neumann entropy and S w ( B |{ P A ( x ) } ) = p (+ x ) S ( ρ B | P A (+ x ) ) + p ( − x ) S ( ρ B | P A ( − x ) ) , (2)with ρ B | P A ( ± x ) the post-weak-measured state given by ρ B | P A ( ± x ) = T r A [( P A ( ± x ) ⊗ I B ) ρ AB ( P A ( ± x ) ⊗ I B )] T r AB [( P A ( ± x ) ⊗ I B ) ρ AB ( P A ( ± x ) ⊗ I B )] , (3)and p ( ± x ) the corresponding probability given by p ( ± x ) = T r AB [( P A ( ± x ) ⊗ I B ) ρ AB ( P A ( ± x ) ⊗ I B )] . (4)It is noted that (cid:26) P (+ x ) = απ + βπ P ( − x ) = βπ + απ , (5)where P † (+ x ) P (+ x ) + P † ( − x ) P ( − x ) = I denotes the two-outcome weakmeasurement with π i the normal projectors, α = r − tanh x , β = r x . (6)and x ∈ R the strength of measurement process. It is obvious that the weakmeasurement operators will be reduced to orthogonal projectors with x → ∞ .The characteristic of the super quantum discord (SD) is that the SDcan catch ‘extra’ quantumness compared with the initial quantum discord.However, it is opposite to the original intention of quantum discord whichrequires to remove the classical correlation from the total correlation as muchas possible by choosing the optimal measurements. It is obvious the weakmeasurement is the bad choice which cannot extract enough classical correla-tion. That is, the super quantum discord should include the residual classicalcorrelation, so it seems to be greater than the usual quantum discord. How-ever, the super quantum discord inspires us to study quantum correlation in3 different way, that is, to be effectively related to the weak-measurementprocess. To do so, we define the quantum correlation cost (QCC) based onthe trace norm as some kind of quantum correlation measure which is givenas follows. Definition 1.
The QCC D w ( ρ AB ) for a bipartite quantum state ρ AB isdefined as D w ( ρ AB ) := min π k ρ AB − Π ( ρ AB ) k , (7)where k X k = T r √ XX † and Π ( · ) denotes the operator of two-outcomeweak measurement on subsystem A withΠ ( ρ AB ) = P (+ x ) ρ AB P † (+ x ) + P ( − x ) ρ AB P † ( − x ) . (8)It is implied that the QCC is defined in terms of the two-outcome weakmeasurement, which holds throughout this paper if no particular statementsgiven. In addition, one can easily find that D w ( ρ AB ) inherits the advantageof the trace norm such as the invariance under local unitary operations andthe contractivity under the local non-unitary evolution on subsystem A , thusit is a reliable measure of quantum correlation. In particular, one should notethat the TQD D ( ρ AB ) for ρ AB can be directly obtained by requiring x → ∞ ,that is, D ( ρ AB ) = lim x →∞ D w ( ρ AB ) . (9)With Eq. (7) and Eq. (9), we can fortunately find the deeper relationbetween the QCC and the TQD which is given by the following rigorousway. Theorem 1.
For a bipartite quantum system of qubits ρ AB D w ( ρ AB ) = (1 − αβ ) D ( ρ AB ) . (10)with α , β defined by Eq. (6) and only determined by the strength of themeasurement process. Proof.
Let’s consider an arbitrary two-qubit state ρ AB . Suppose thatthe final state after the weak measurement is Π ( ρ AB ). Substitute Eq. (8)into Eq. (7), we have D w ( ρ AB ) = min π k ρ AB − Π ( ρ AB ) k = min π k ρ AB − ( π ρ AB π + π ρ AB π ) − αβ ( π ρ AB π + π ρ AB π ) k = (1 − αβ ) min π k π ρ AB π + π ρ AB π k , (11)4nsert Eq. (9) into Eq. (11), it follows that D w ( ρ AB ) = (1 − αβ ) D ( ρ AB ) . (12)which completes the proof. (cid:4) Theorem 1 shows us a very simple factorization relation between the QCCand the TQD, which provides the important root for the next stories.
Residual quantumness.-
Based on the factorization relation given above,one will obviously see that D w ( ρ AB ) ≤ D ( ρ AB ) due to the reduction factor(1 − αβ ) ≤ x → ∞ . Since the weak measurementinfluences the system more weakly than the normal projective measurement,the distance from the state of interests to the post-weak-measured stateis naturally less than that from the state to the post-projective-measuredstate. Therefore, compared with the TQD D ( ρ AB ) , it is shown that theQCC D w ( ρ AB ) can only grasp the partial quantumness instead of the extraquantumness. Thus the residual quantumness can be written as∆ = D ( ρ AB ) − D w ( ρ AB ) = 2 αβD ( ρ AB ) . (13)Below we will show that the residual quantumness can be further extracted ifwe continue performing a projective measurement on the post-weak-measuredstate. Theorem 2.
Let ˜ ρ AB denote the final state of ρ AB after the optimalweak measurement such that D w ( ρ AB ) = k ρ AB − ˜ ρ AB k , then we have D ( ˜ ρ AB ) = D ( ρ AB ) − D w ( ρ AB ) . (14) Proof.
At first, one should keep in mind that ˜ ρ AB = Π ( ρ AB ) given byEq. (8) can be further written as that implied in Eq. (11) based on the Proofof Theorem 1. In addition, in order to distinguish the projectors from thosein the weak measurement, we denote the projectors operated on subsystem A in the normal projective measurement by be π ′ and π ′ . Therefore, in termsof the definition of TQD, one can obtain D ( ˜ ρ AB ) as D ( ˜ ρ AB ) = min π ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ ρ AB − X i =1 π ′ i ˜ ρ AB π ′ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (15)Since for x → ∞ , the weak measurement operated on ˜ ρ AB should become5he normal projective measurement, one will arrive atlim x →∞ D ( ˜ ρ AB ) = min π ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j =1 π j ρ AB π j − X i,j =1 π ′ i π j ρ AB π j π ′ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 . (16)where π and π are the projectors in the weak measurement. It is obviousthat the equality in Eq. (16) holds iff the set of the projectors { π ′ i } = { π j } .Thus the extremum operation in Eq. (16) can be omitted, and then Eq. (16)can be rewritten as D ( ˜ ρ AB ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ ρ AB − X i =1 π i ˜ ρ AB π i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (17)Substitute the expression of ˜ ρ AB into Eq. (17), we will arrive at D ( ˜ ρ AB ) = k π ˜ ρ AB π + π ˜ ρ AB π k = 2 αβ k π ρ AB π + π ρ AB π k . (18)where we use π i π j = δ ij π i . The weak measurement in ˜ ρ AB is required tobe optimal in the sense of D w ( ρ AB ) = k ρ AB − ˜ ρ AB k , so the projectors inthe weak measurement will also be optimal for the corresponding TQD for x → ∞ , because if it is not the case, it will lead to two different TQDs forthe same ρ AB , which is also interpreted in Ref. [50]. That is, D ( ρ AB ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ AB − X i =1 π i ρ AB π i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k π ρ AB π + π ρ AB π k . (19)Eq. (18) and Eq. (19) show that D ( ˜ ρ AB ) = 2 αβD ( ρ AB ) . (20)which accompanied with Theorem 1 implies D ( ˜ ρ AB ) = D ( ρ AB ) − D w ( ρ AB ).The proof is finished. (cid:4) In Theorem 2, we have shown that the residual quantum correlation thatthe weak measurement fails to ‘extract’ can be ‘extracted’ by the latter pro-jective measurement. In addition, it is implied that no extra quantumness is6asted compared with one optimal projective measurement, even though weuse weak measurements and projective measurements, respectively. That is,the summation of the QCC and the residual TQD is completely consistentwith the TQD. Since any projective measurement can be implemented by asequence of continuous weak measurements, one could naturally ask if thequantum correlation can be extracted little by little by these weak measure-ments. In the following theorem, we will show that the weak measurementcan do this job indeed.
Corollary.
Suppose ρ n to be the final state after n optimal weak mea-surements with the same infinitesimal measurement strength on the subsys-tem A of ρ AB such that D w ( ρ n ) = k ρ n − ρ n +1 k with ρ = ρ AB , then we willhave D w ( ρ n ) = (1 − αβ ) (2 αβ ) n D ( ρ AB ) , (21)and D ( ρ AB ) = lim x → ∞ X n =0 D w ( ρ n ) . (22) Proof.
According to Theorem 1, for ρ n we have D w ( ρ n ) = (1 − αβ ) D ( ρ n ) . (23)Using Theorem 2 and its proof, one can find D ( ρ n ) = D ( ρ n − ) − D w ( ρ n − ) = 2 αβD ( ρ n − ) , (24)which directly leads to D ( ρ n ) = (2 αβ ) n D ( ρ AB ) . (25)Insert Eq. (25) into Eq. (23), we will obtain D w ( ρ n ) = (1 − αβ ) (2 αβ ) n D ( ρ AB ) . (26)Since we consider the infinitesimal measurement strength which means x →
0, i.e., αβ →
0, sum Eq. (26) over n , one will getlim x → ∞ X n =0 D w ( ρ n ) = lim x → (1 − αβ ) D ( ρ AB ) ∞ X n =0 (2 αβ ) n = D ( ρ AB ) . (27)7q. (26) and Eq. (27) complete the proof. (cid:4) Since only partial quantum correlation can be grasped by the weak mea-surement, the above corollary first shows that the quantum correlation canbe ‘extracted’ continuously by the infinitesimal weak measurements, whichis consistent with the fact that the continuous infinitesimal weak measure-ments can realize the projective measurement. What is important is thatthe series of weak measurements do not waste extra quantumness either. Onthe contrary, if consider that the projective measurement will destroy all thequantum correlation, in principle after the projective measurement, the lat-ter weak measurement will ‘extract’ no quantum correlation. In the followingtheorem, we will prove that it is the case.
Theorem 3.
Let the final state of ρ AB after any projective measurementon subsystem A is given by ˜ ρ ′ AB = P j =1 ˜ π ′ j ρ AB ˜ π ′ j , then we have D w ( ˜ ρ ′ AB ) = 0 . (28) Proof.
Since ˜ ρ ′ AB = P j =1 ˜ π ′ j ρ AB ˜ π ′ j , it is obvious that D ( ˜ ρ ′ AB ) = 0. Ac-cording to Theorem 1, one can easily find that D w ( ˜ ρ ′ AB ) = (1 − αβ ) D ( ˜ ρ ′ AB ) =0. The proof is completed. (cid:4) The case of multiple-outcome weak measurement.-
One can find that allthe jobs presented above only cover the two-outcome weak measurement.Next we will extend the weak measurement to the case of n outcomes whichcan be written as P ( i ) = α i π + β i π , (29)where the real α i , β i with P ni =1 α i = 1 and P ni =1 β i = 1 are determined bythe strength vector ~x of measurement process and similarly π and π arethe projectors satisfying π + π = I . In particular, one should note thatin the limitation ~x → ∞ , P ( i ) will become the normal projective measure-ment. Following the same procedure as we’ve done for the two-outcome weakmeasurement, we will have the following theorem. Theorem. 4 . All the above conclusions hold for n -outcome weak mea-surement if the factor 2 αβ corresponding to two outcomes is replaced by P ni =1 α i β i corresponding to the n outcomes. Proof.
The proof is completely analogous to those given for the two-outcome case, so it is omitted. (cid:4) (2 ⊗ d ) -dimensional systems.- Finally, we would like to emphasize that allthe theorems as well as the corollary hold for (2 ⊗ d )-dimensional systems,8hich can be confirmed if following the completely analogous proof procedurefor the two-outcome case.To sum up, one can easily find that the QCC is quite different from theSD as well as the TQD. That is, the QCC characterizes how much quantumcorrelation is ’extracted’ (used) in the weak-measurement process. Eventhough the quantum correlation in a system can be ’extracted’ by infiniteweak measurements, the QCC subject to the weak measurement per se doesnot describe how much correlation is present in the system. As mentionedpreviously, the SD badly characterizes the classical correlation in a system,which can be seen when the measurement strength tends to zero ( the SDwill reach its maximum, the total correlation and the QCC will reach its min-imum, zero). The TQD describes the quantum correlation in the consideredsystem.
3. The application
As a demonstration, let’s consider the Bell-diagonal states given by ρ AB = 14 ( I AB + X k =1 c k σ Ak ⊗ σ Bk ) , (30)with σ k the Pauli matrices and | c | > | c | > | c | . Based on Refs. [26, 28, 37],one can easily find that the QCC and TQD are given by D ( ρ AB ) = | c | , (31)and D w ( ρ AB ) = (1 − αβ ) | c | . (32)In particular, one can find that the optimal projectors that lead to the QCCand TQD are ˆ π = | i h | and ˆ π = | i h | . Eq. (31) and Eq. (32) show theconsistency with Theorem 1. Substitute the optimal projectors into the weakmeasurement, one can obtain the weak measurement as ˆ P ( x ) = α ˆ π + β ˆ π and ˆ P ( − x ) = β ˆ π + α ˆ π . So we can write the post-measured state as˜ ρ AB = ˆ P ( x ) ρ AB ˆ P ( x ) + ˆ P ( − x ) ρ AB ˆ P ( − x )= 14 ( I AB + X k =1 ˜ c k σ Ak ⊗ σ Bk ) , (33)9ith ˜ c k = 2 αβc k for k = 1 , c = c . Use Eq. (18) again, one can easilyfind that D ( ˜ ρ AB ) = 2 αβ | c | , (34)which associated with Eqs. (31,32) demonstrates the validity of Theorem2. In addition, if we continue performing a weak measurement on ˜ ρ AB , onewill find that ˜ c k will be reduced further by a factor 2 αβ and c will bestill invariant. So one will find D ( ˜ ρ ) = (2 αβ ) | c | and D w ( ˜ ρ ) = (1 − αβ ) (2 αβ ) | c | with ˜ ρ representing the twice weak measurements. Thisis consistent with our corollary. Finally, let the projectors ˆ π and ˆ π beperformed on the subsystem A of ρ AB , one will get the final state as˜ ρ ′ AB = ˆ π ρ AB ˆ π + ˆ π ρ AB ˆ π = 14 (cid:2) | i h | ⊗ (cid:0) I B + c σ B (cid:1) + | i h | ⊗ (cid:0) I B − c σ B (cid:1)(cid:3) . (35)which obviously has no quantum correlation. The similar conclusion for ˜ ρ ′ AB can also be found if one employs arbitrary projectors instead of ˆ π and ˆ π .Therefore, Theorem 3 in this Bell-diagonal state is also satisfied.
4. Conclusion and discussions
We have presented the QCC for the weak measurement based on thetrace norm. The QCC quantifies the quantumness ‘extracted’ by the em-ployed weak measurements during the measurement procedure. We havefound a factorization relation between the QCC and the TQD. This is easilyunderstood since weak measurement only influences the system weakly. Itis especially interesting that the residual quantumness after the weak mea-surement can be ‘extracted’ further by the latter projective measurement orby the latter sequence of infinitesimal weak measurements, which shows theconsistent nature with that the sequence of infinitesimal weak measurementscan realize the projective measurements. It is important that these differ-ent measurements do not waste extra quantumness. In contrast, it is shownthat the weak measurement cannot extract any quantum correlation fromthe state after projective measurements. We believe that it provides a newpoint of view for us to understand the weak measurement and the projectivemeasurement. In addition, we also generalize our conclusions to the cases of2 ⊗ d -dimensional systems and of the multiple-outcome measurement. Finally,we demonstrate our theorems and corollary by the Bell-diagonal states.10 cknowledgement This work was supported by the National Natural Science Foundation ofChina, under Grants No.11375036 and 11175033.
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