Dynamics of Super Quantum Correlations and Quantum Correlations for a System of Three Qubits

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1 Commun. Theor. Phys. 65 (016) Vol. 65 No. 4 April Dynamics of Super Quantum Correlations and Quantum Correlations for a System of Three Qubits F. Siyouri M. El Baz S. Rfifi and Y. Hassouni Laboratoire de Physique Thèorique Département de Physique Faculté des sciences Université Mohammed V Agdal Av. Ibn Battouta B.P Agdal Rabat Morocco (Received October 1 015; revised manuscript received December ) Abstract The dynamics of quantum discord for two qubits independently interacting with dephasing reservoirs have been studied recently. The authors [Phys. Rev. A (013) ] found that for some Bell-diagonal states (BDS) which interact with their environments the calculation of quantum discord could experience a sudden transition in its dynamics this phenomenon is known as the sudden change. Here in the present paper we analyze the dynamics of normal quantum discord and super quantum discord for tripartite Bell-diagonal states independently interacting with dephasing reservoirs. Then we find that basis change does not necessary mean sudden change of quantum correlations. PACS numbers: Ta Yz Mn Key words: super quantum discord normal quantum discord Bell-diagonal states measurement strength 1 Introduction In recent years the quantification and investigation of quantum correlations which are not limited to the entanglement measures [1] have been one of the most widely studied topics of quantum information science. Moreover there are other quantum correlations existing in separable quantum states that can have quantum behavior and offer faster computational in quantum information processes. [] Among many measures which have been proposed to quantify these quantum correlations quantum discord [3] confirmed that it is the most popular one and may play an important role and more advantages in implementing quantum information tasks. On the other hand the measurement of an arbitrary quantum state in some orthogonal basis (proective measurement) leads to the loss of its coherence. However the system will be perturbed slowly and may not lose its coherence completely when we perform a measurement that couples the system and the measuring device weakly. [4] In fact Aharonov Albert and Vaidman introduced weak measurements [1] which have disputed the idea that the value of an observable in quantum mechanics has physical reality only if it is actually measured. Weak measurements are used for interrogating quantum systems by helping to understand the macrorealism [5 6] and the role of the uncertainty principle in the double-slit experiment [7 ] also they are used by Hosten and Kwiat [9] to observe the spin Hall effect in light by Dixon et al. [10] to detect very small transverse optical beam deflections and by Gillett et al. [11] to examine the feedback control of quantum systems in the presence of noise and direct measurements of the wave function of a single photon. [1] Recently the weak measurements were shown to help even in the protection of quantum entanglement from decoherence. [13] They are important for exploring the extra quantum correlation for information processing. [14] Also they have been proven to reveal more quantum correlations for a bipartite quantum system. Thus a weak measurement performed on one of the subsystems can lead to super quantum discord which is always larger than the normal quantum discord captured by strong measurements (proective). Also the authors of Ref. [1] have proven that super quantum discord is a monotonic function of the measurement strength and that it covers all values between mutual information and normal quantum discord. In order to understand the dynamics of normal quantum discord and super quantum discord it is important to consider the quantum system as open. In our case the open quantum system was independently interacting with dephasing reservoirs. Only a few studies have treated the effect of the environment on quantum discord and super quantum discord. Therefore the investigation of processing and application of quantum correlations in open quantum systems become both practically important and fundamental. Lately Pinto et al. [15] found that for bipartite of some Bell-diagonal states that interact with their environments the calculation of quantum discord could experience a sudden transition in its dynamics therefore the time derivative of quantum discord might become intermittent at some points during the time evolution of the system. In addition to that the time evolution of quantum discord could freeze and let it evolved independently of the destructive effects of the open system. [16 17] In this work we extend our study to the case of a three qubits fatimazahra.siyouri@gmail.com elbaz@fsr.ac.ma saad.rfifi@gmail.com y-hassou@fsr.ac.ma c 016 Chinese Physical Society and IOP Publishing Ltd

2 44 Communications in Theoretical Physics Vol. 65 of Bell-diagonal states to analyze the dynamics of quantum correlations in open systems under a specific type of non-markovian noise. The article is divided as follows. In Sec. we briefly discuss our model of interest and the quantum states which we study in this work. In Sec. 3 we review the theoretical development of quantum discord for tripartite systems also we use the weak measurements which lead to calculate the super quantum discord for the same quantum system. That allows to control the amount of quantum correlation in terms of the measurement strength according to our need in the quantum information process (QIP). We provide results in Sec. 4. And finally we conclude our work in Sec 5. Non Markovian Dephasing Model for Bell Diagonal States The arbitrary bipartite Bell-diagonal states have been widely used in the literature to demonstrate the sudden change of quantum discord in open quantum systems. [17 19] In our case we are analyzing the sudden change of quantum discord and super quantum discord for tripartite Bell-diagonal states whose reduced states are degenerate namely ρ(0) = I 3 + c σ 1 σ σ3 (1) =1 where c are real numbers such that 0 c 1 and I is the identity matrix. The eigenvalues of initial density matrix ρ(0) are ( ) λ 134 = 1 1 ( 1 + c 1 + c + c 3 c 1 + c + c 3 ). () λ 567 = 1 We consider a colored noise dephasing model with dynamics described by a master equation [150] ρ = KLρ (3) where K is a time-dependent integral operator whose action on the system is defined as Kφ = t 0 k(t t)φ( t)d t (4) with k(t t) is a kernel function which determines the type of memory in the environment ρ is the density matrix of the principal system and L is the Lindblad superoperator which describes the open system dynamics as a result of the interaction between the environment and the principal system. One usually obtains the master equation with Markovian approximation even in the absence of K in Eq. (3). If we consider a master equation as a two-level quantum system that interacts with a reservoir having the properties of random telegraph signal noise this type of master equation may arise. To analyze it we can begin with a time-dependent Hamiltonian. [150] 3 H(t) = Γ k (t)σ k (5) k=1 where σ k are the usual Pauli matrices and Γ k (t) are independent random variables which obey the statistics of a random telegraph signal. In particular the random variables can be expressed as Γ k (t) = a k n k (t) where n k (t) has a poisson distribution with a mean equal to t/τ k and a k is an independent random variable taking values ±a k. Using von Neumann equation of motion ρ = (i/ )[H ρ] to get a solution for the density matrix of the two-level system having the form ρ(t) = ρ(0) i t 0 Γ k (s)[σ k ρ(s)]ds. (6) k We substitute this equation back into von Neumann equation and we perform stochastic average we get [0] ρ = t 0 exp( t t /τk )a k[σ k [σ k ρ( t)]]d t. (7) k The memory kernel obtained from the correlation functions of random telegraph signal is given by: Γ (t)γ k ( t) = a k exp( t t /τk )δ k. () According to some recently introduced measures of non-markovianity [1 ] the model described above gives rise to a non-markovian time evolution. It seems that the dynamical evolution generated by Eq. (7) is completely positive when two of the a k are zero. This corresponds to a physical situation where noise only acts in one direction. Particularly if a 1 = a = 0 and a 3 = a the dynamics of the system is that of a dephasing channel with colored noise. Consequently the Kraus operators describing the dynamics of two-level system are given by [150] 1 + Λ(ν) K 1 = I (9) 1 Λ(ν) K = σ 3 (10) where I is the identity matrix and the Kraus operators satisfy the normalization condition i K i K i = I. With Λ(ν) = e ν [cos(µν)+sin(µν)/µ] µ = (4aτ) 1 and ν = t/τ is the dimensionless time. As we are interested in three-qubit system the time evolution of an initial density matrix can be written as ρ(t) = i M i (t)ρ(0)m i (t) (11) where ρ(0) is the initial state of the three-qubit system and K i (t) are the Kraus operators satisfying the normalization condition i M i (t)m i(t) = I. For three qubits there are such operators that is M 1 = K A 1 K B 1 K C 1 M = K A 1 K B 1 K C M = K A K B K C. The density matrix after the damping has the following

3 No. 4 Communications in Theoretical Physics 449 form with a b 0 d c d 0 0 c 0 0 ρ 13 = a b c d b 0 0 a b a 0 c d a = 1 + c 3 b = (c 1 + ic )Λ(ν) 3 d = 1 c 3 c = (c 1 ic )Λ(ν) 3. Then the eigenvalues of the density matrix are ( ) c 1 + (c + c3 )Λ(ν)6 λ 134 = 1 1 ( 1 + c 1 + (c + c3 )Λ(ν)6 ). (1) λ 567 = 1 Now we turn our attention to normal quantum discord and super quantum discord after introducing the definition of the non-markovian dephasing model which we intend to study in our investigation. 3 Normal Quantum Correlations and Super Quantum Correlations 3.1 Normal Quantum Discord (NQD): Quantum discord measures the amount of information that cannot be obtained by performing a measurement on one subsystem alone so it can be considered as a measure of quantum correlations. Indeed many works have tackled the problem of quantifying quantum correlations in various types of bipartite states. [3 4] Thus extending the study of quantum discord from bipartite to tripartite systems. [5 6] In particular Fanchini et al. [7] are introduced quantum discord for three qubits Rulli et al. [] have showed that even completely separable mixed states non-zero quantum discord was sufficient to teleport the quantum information. Let us consider a two random variables X and Y. The two equivalent expressions of mutual information between the variables defined by: I(X : Y ) = H(X) + H(Y ) H(X Y ) (13) J(X : Y ) = H(X) H(X Y ). (14) Here H(X) H(Y ) and H(X Y ) are Shannon entropies [3] for the random variables X and Y and the pair (X Y ) respectively and H(X Y ) is the conditional entropy. [3] The quantum analog [9] of these quantities is given as: I(X : Y ) = S(ρ X ) + S(ρ Y ) S(ρ XY ) (15) J(X : Y ) {Π Y } = S(ρ X ) S(ρ X {Π Y }). (16) In this case X and Y stand for quantum subsystems. S(ρ X ) S(ρ Y ) and S(ρ XY ) [30] are respectively the von Neumann entropies for the quantum subsystems X Y and the oint X and Y. The parameter {Π Y } represents the measurement basis for subsystem Y and S(ρ X {Π Y }) is the conditional entropy [9] for subsystem X when the complete measurement over subsystem Y is performed and it can be expressed as with S(ρ X {Π Y }) = p = Tr XY (I Π Y ρ XY ) p S(ρ X Π Y ) (17) ρ X Π Y = 1 p (I Π Y ρ XY I Π Y ). (1) The quantum quantities in Eqs. (15) and (16) turn out to be not equivalent in contrast to the classical case. In order to ensure that this last definition takes into account all classical correlations one must maximize it over all possible measurement basis {Π Y } of subsystem Y.[9] Consequently the quantum discord capturing all quantum correlations is defined as: ] D(X : Y ) {Π Y } = I(X : Y ) max {Π Y }[ J(X : Y ){Π Y = } S(ρY ) S(ρ XY ) + min {Π Y }S ( ) ρ X {Π Y }. (19) It turns out that maximizing Eq. (16) is the main difficulty in finding general analytic expressions for quantum discord present in arbitrary states. Indeed exact analytical expressions are found only in a limited number of cases and the most general approach up to date was obtained for the so-called X-states. [31] This approach has been successfully applied to understand and quantify the quantum correlations present in different types of systems. [3 34] 3. Super Quantum Discord (SQD): After defining normal quantum discord with strong measurements let us spend to the super quantum discord. The weak measurement operators are given by [35] 1 tanhx 1 ± tanhx P(±x) = Π 0 + Π 1 (0) with P (x)p(x) + P ( x)p( x) = I where x is a parameter that denotes the strength of the measurement process Π 0 and Π 1 are two orthogonal proectors that satisfy Π 0 + Π 1 = I. In addition lim P( x) = Π 0 and x + lim P(x) = Π 1. x + Thus the super quantum discord denoted by D w (X : Y ) is defined as D w (X : Y ) = S(ρ Y ) S(ρ XY ) + min {P Y (x)} [S w (X {P Y (x)})] where the weak quantum conditional entropy is given by S w (ρ X {P Y (x)}) = P(x)S w (ρ X P Y (x)) + P( x)s w (ρ X P Y ( x)) (1)

4 450 Communications in Theoretical Physics Vol. 65 with ρ X P Y (±x)= Tr Y [(I P Y (±x))ρ XY (I P Y (±x))] Tr XY [(I P Y (±x))ρ XY (I P Y (±x))] () and P(±x) = Tr XY [(I P Y (±x))ρ XY (I P Y (±x))]. (3) The terms in the super quantum discord equations Eq. (1) are the same as terms in the normal quantum discord equations Eq. (19) except for the conditional entropy of the weak measurement term. Thus in this section we calculate the weak quantum conditional entropy Eq. (1) to use instead of Eq. (17). For this the two weak measurement proectors are ( ) χ ν + ϕ Γ η ν η Γ P 3 ( x) = κ ν κ Γ ϕ ν + χ Γ ( ) ϕ ν + χ Γ η ν + η Γ P 3 (+x) = κ ν + κ Γ χ ν + ϕ Γ where Γ = (1 + tanh(x))/ ν = (1 tanh(x))/ η = cosθ sin θ e iφ κ = cosθ sin θ e iφ ϕ = sin θ and χ = cos θ. The two appropriate probabilities are defined as Prob(±x)=Tr[(I I P 3 (±x))ρ 13 (I I P 3 (±x))].(4) After calculating Tr 3 [(I I P 3 (±x))ρ 13 (I I P 3 (±x))] (5) using Eq. () which allows the density matrix of each weak measurement to be extracted we find that ρ(1 P 3 (±x)) = 1 Prob(±x) Tr 3[(I I P 3 (±x))ρ 13 (I I P 3 (±x))]. (6) Then by using Eq. (1) we can easily deduce the expression S w (ρ 1 {P3 (±x)}). Thus the super quantum discord Eq. (1) of the studied system is easy found to be D w (X : Y ) = S(ρ 3 ) S(ρ 13 ) + min {P 3 (x)}[s w (1 {P 3 (x)})]. (7) 4 Results In order to explore the effects of decreasing the degree of non-markovianity on its dynamics for various values of parameter τ we plotted the dynamics of normal quantum discord of BDS for a three qubits system interacting with independent colored dephasing reservoirs against parameter t/τ in Fig. 1. Note that by fixing a = 1 s the parameter τ = 5 s denotes the non-markovian case whereas τ = 1 s corresponds to the Markovian case. We plot in Fig. the dynamics of super quantum discord for a three qubits system interacting with independent colored dephasing reservoirs. Then we analyze the transitions between different measurement bases during the time evolution of the system. NQD a/1 s t=5 s t=4 s t=3 s t= s t=1 s t t Fig. 1 Dynamics of NQD for the BDS described by the parameters c 1 = 1 c = 0.6 and c 3 = 0.6 as a function of t/τ. Fig. Dynamics of SQD for the BDS described by the parameters c 1 = 1 c = 0.6 and c 3 = 0.6 as a function of t/τ. In Fig. 3 we plot the minimizing basis for the BDS described by the parameters c 1 = 1 c = 0.6 and c 3 = 0.6 as a function of t/τ. Figure 1 shows that the evolution behavior of NQD in non-markovian regime differs essentially from that in Markovian regime. Moreover it shows that when we decrease the degree of non-markovianity τ the collapses and revivals of NQD are also decreased and delayed.

5 No. 4 Communications in Theoretical Physics 451 In non-markovian regime (τ > ) due to the environmental memory effect which allows an increase in the information backflow [36] the NQD decreases gradually to zero and then revives after a period of time with a damping amplitude. While in Markovian regime (τ < ) a weak system-environment coupling and a memoryless transfer of information from the system to the environment leading to the suppression of information [37] so the NQD decreases asymptotically to zero without any revival (the system continuously loses information to the environment). This Markovian behavior is consistent with the results of GHZ and W states that are obtained in Refs. [3-39]. Also we know that the tripartite BDS initially have no interaction with the open system so the revival phenomenon is due to single qubit non-markovian dynamics resulting from the feed-back effect of environment. Furthermore (τ) decreases with increasing of the Markovianity as also the revival amplitude (more information may be returned to the system from the environment). Thus it can be seen from Fig. that the SQD attains the maximum value at x = 0 where the weak measurement is the weakest. When x 4.5 the SQD approaches NQD. We observe that the dynamics of SQD value decreases monotonically with the increasing measurement strength parameter x. The behavior of the super quantum correlation and the quantum correlation keep unchanged for the tripartite Bell-diagonal states. On the other hand it is remarkable that no sudden change appears between different measurements bases during the time evolution of the system in tripartite generation of BDS either for NQD or SQD. Figure 3 shows that during the time evolution of the system in tripartite generation of BDS the measurement basis changes from π/4 to π then back to π/. Furthermore the basis change rate depends on the amount of non-markovianity (τ). For large value of τ the value of µ is large so the cosine and sine functions are considerable and cannot be ignored. While for small value of τ the value of µ is small consequently we can ignored the the cosine and sine functions. Hence these functions are responsible for the collapses and revivals as well as the basis change rate during the time evolution of the system. From Fig. 1 Fig. and Fig. 3 we can conclude that basis change does not necessary mean sudden change of quantum correlations and super quantum correlations for tripartite BDS i.e. it is a necessary condition but not a sufficient one. Fig. 3 Minimizing basis for the BDS described by the parameters c 1 = 1 c = 0.6 and c 3 = 0.6 as a function of t/τ. 5 Conclusion In this report we have studied the dynamics of NQD and SQD for a tripartite BDS interacting with independent dephasing reservoirs we have first shown that the evolution of their behavior in non-markovian regime differs essentially from that in Markovian regime. However it can be observed that as we decrease the degree of non- Markovianity τ the collapses and revivals of NQD and SQD are also decreased and delayed. Also the dynamics of SQD value decreases monotonically with the increasing measurement strength parameter x and it can approach NQD when x. Furthermore the dynamical behavior of quantum correlations depends only on the parameters τ and t/τ. On the other hand by performing weak

6 45 Communications in Theoretical Physics Vol. 65 measurements on the access of the same quantum system it will be clear that the dynamical behavior of super quantum correlations additionally will depend on the measurement strength x. Moreover we have also tested the possibility of occurrence of sudden change for three qubits BDS and we have found that despite the appearance of basis change between different measurement bases during the time evolution of the system no sign of such a dynamical behavior appears either for NQD or SQD. So basis change does not necessary mean sudden change of quantum correlations and super quantum correlations for tripartite BDS i.e. it is a necessary condition but not a sufficient one. References [1] Y. Aharonov D.Z. Albert and L. Vaidman Phys. Rev. Lett. 60 (19) [] A. Datta A. Shai and C.M. Caves Phys. Rev. Lett. 100 (00) [3] H. Ollivier and W.H. Zurek Phys. Rev. Lett. (001) [4] L. Wang J.H. Huang J.P. Dowling and S.Y. Zhu Quantum Inf. Proc. 1 (013) 99. [5] Y. Aharonov S. Popescu and J. Tollaksen Phys. Today 63 (010) 7. [6] N.S. Williams and A.N. Jordan Phys. Rev. Lett. 100 (00) [7] A. Palacios-Laloy et al. Nature Phys. 6 (010) 44. [] H.M. Wiseman Phys. Lett. A 311 (003) 5. [9] O. Hosten and P. Kwiat Science 319 (00) 77. [10] D.J. Starling et al. Phys. Rev. A 0 (009) 04103(R). [11] N. Brunner and C. Simon Phys. Rev. Lett. 105 (010) [1] G.A. Smith et al. Phys. Rev. Lett. 93 (004) [13] J.S. Lundeen et al. Nature (London) 474 (011) 1. [14] Y.S. Kim et al. Nat. Phys. (01) 117. [15] Joo P.G. Pinto Gktu Karpat and Felipe F. Fanchini Phys. Rev. A (013) [16] L. Mazzola J. Piilo and S. Maniscalco Phys. Rev. Lett. 104 (010) [17] L. Mazzola J. Piilo and S. Maniscalco Int. J. Quantum Inf. 09 (011) 91. [1] J. Maziero L. C. Cleri R. M. Serra and V. Vedral Phys. Rev. A 0 (009) [19] B. You and L.X. Cen Phys. Rev. A 6 (01) [0] S. Daffer K. Wodkiewicz J.D. Cresser and J.K. McIver Phys. Rev. A 70 (004) [1] S. Luo S. Fu and H. Song Phys. Rev. A 6 (01) [] H.P. Breuer E.M. Laine J. Piilo Phys. Rev. Lett. 103 (009) [3] M.A. Nielsen and I.L. Chuang Cambridge University Press Cambridge (000). [4] I.A. Silva D. Girolami et al. Phys. Rev. Lett. 110 (013) [5] B.L. Ye Y.M. Liu X.S. Liu and Z.J. Zhang Chin. Phys. Lett. 30 (013) [6] I. Chakrabarty P. Agrawal and A.K. Pati Euro. Phys. J. D 65 (011) 605. [7] F.F. Fanchini M.F. Cornelio M.C. de Oliveira and A.O. Caldeira Phys. Rev. A 4 (011) [] C.C. Rulli and M.S. Sarandy Phys. Rev. A 4 (011) [9] L. Henderson and V. Vedral J. Phys. A: Math. Gen. 34 (001) 699. [30] T.M. Cover and J.A. Thomas Elements of Information Theory Ed. J. Wiley New York (1991). [31] Ali Mazhar A.R.P. Rau and G. Alber Phys. Rev. A 1 (010) [3] F.F. Fanchini L.K. Castelano and A.O. Caldeira New J. Phys. 1 (010) [33] M.F. Cornelio M.C. de Oliveira and F.F. Fanchini Phys. Rev. Lett. 107 (011) [34] M. Shi W. Yang F. Jiang and J. Du J. Phys. A: Math. Theor. 44 (011) ; J.Xu J. Phys. A: Math. Theor. 44 (011) [35] O. Oreshkov and T.A. Brun Phys. Rev. Lett. 95 (005) [36] C. Addis P. Haikka S. McEndoo C. Macchiavello and S.Maniscalco Phys. Rev. A 7 (013) 05109; S. Haseli G. Karpat S. Salimi A.S. Khorashad F.F. Fanchini B. Cakmak G.H. Aguilar S.P. Walborn and P.H. Souto Ribeiro Phys. Rev. A 90 (014) [37] V. Gorini A. Frigerio M. Verri A. Kossakowski and E.C.G. Sudarshan Rep. Math. Phys. 13 (197) 149. [3] M. Mahdian R. Yousefanib and S. Salimi Eur. Phys. J. D 66 (01) 133. [39] H. Guo J. Liu C. Zhang and C.H. Oh Quantum Inf. Comput. 1 (01) 677.