The dynamics of quantum correlation with two controlled qubits under classical dephasing environment

The dynamics of quantum correlation with two controlled qubits under classical dephasing environment

Accepted Manuscript The dynamics of quantum correlation with two controlled qubits under classical dephasing environment Jun-Qi Li, Xin-Lin Cui, J.-Q...

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Accepted Manuscript The dynamics of quantum correlation with two controlled qubits under classical dephasing environment Jun-Qi Li, Xin-Lin Cui, J.-Q. Liang PII: DOI: Reference:

S0003-4916(15)00008-1 http://dx.doi.org/10.1016/j.aop.2015.01.005 YAPHY 66704

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Annals of Physics

Received date: 22 September 2014 Accepted date: 6 January 2015 Please cite this article as: J.-Q. Li, X.-L. Cui, J.-Q. Liang, The dynamics of quantum correlation with two controlled qubits under classical dephasing environment, Annals of Physics (2015), http://dx.doi.org/10.1016/j.aop.2015.01.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The dynamics of quantum correlation with two controlled qubits under classical dephasing environment Jun-Qi Li∗ , Xin-Lin Cui, and J.-Q. Liang Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, Shanxi, China We discuss the quantum correlation dynamics of two qubits controlled through the application of π-pulses under classical dephasing non-Markovian environment. It is shown that the quantum discord (QD) and one-norm geometric quantum discord (one-norm GQD) between the two qubits, which are prepared initially in the three-parameter-X-type quantum states, depend strongly on nonMarkovian properties and the time difference between adjacent pulses. The freezing time of discord and one-norm GQD can be lengthened for appropriate pulse separate time and pulse numbers. And the freezing time of one-norm GQD is longer slightly than QD for both Markovian and nonMarkovian cases. What’s more, we find that double sudden changes of one-norm GQD can appear only for some initial parameters when π-pulses are applied. PACS numbers: 03.67.-a, 03.65.Yz, 03.65.Ta Keywords: quantum correlation; dynamics; environment

I.

INTRODUCTION

Quantum entanglement, a hallmark of quantum correlation, has attracted much attention during recent years in information science and condensed matter physics due to its fundamental importance [1, 2]. However, recent studies [3, 4] show that entanglement is not the only type of correlation useful for quantum technology. There exists other kind of nonclassical correlation other than entanglement that can speed up some computational tasks compared to classical computation [5]. Such correlation may be quantified by new measures of quantum correlation [6–10], among which, QD [6] with a definition based on the distinction between the quantum and classical information theory has received a great deal of interest [11–13]. QD becomes distinguishable from entanglement for mixed states. For instance, discord is nonzero in some mixed separable states yet no entanglement [14]. This suggests that QD is more general than entanglement for capturing the quantumness of correlation. Recently, [15] showed that the QD is also a physical quantity, because erasure of quantum correlation must lead to entropy production in the system and the environment. Moreover, QD is intimately related to many important protocols, such as the distribution of entanglement [16], entanglement irreversibility [17], quantum state merging [18, 19] and quantum metrology [20], etc. Although discord is a useful corrrelation measure, it has analytic expressions known only for certain classes of states [21–23]. To overcome this difficult, [9] proposed the concept of geometric quantum discord (GQD), which is analytical for general two-qubit states [24] as well as for arbitrary bipartite states [25–27] and has been discussed in a wider context [28–32]. Unfortunately, it was recently pointed out [33], GQD cannot be regarded as a good measure for the quantumness of correlation, since it may increase under local operations on the unmeasured subsystem. Nevertheless, this drawback can be completely overcome by the novel measurement of one-norm GQD [34]. The quantum advantage beyond classical properties relies on quantum correlation. However, because of the unavoidable interaction between a quantum system and its environment, quantum correlation may be damaged by noises, which leads to decoherence. Therefore, studying the time-evolving behaviors of quantum correlation is pivotal to its exploitation for real-world implementation of quantum protocols. So far, quantum correlation dynamics in open systems has been a hot topic and discussed extensively by a number of authors in inertial [35–40] or non-inertial frame [41–43]. It has been shown that QD does not exhibit the phenomenon of sudden death [35] and is more robust than entanglement under the Markovian and non-Markovian environments [44–46]. Especially, the phenomenon of single sudden transition of discord was reported theoretically by [47] in Markovian noise for some specific initial conditions. This sudden transition implies that there is a critical time tc before which QD is ”frozen” to its initial value while the classical correlation decays with time, then after which QD starts to decays while classical correlation is ”frozen”. This peculiar behavior that never observed before has intrigued extensively research interest [48–51] and verified experimentally [52–55]. More interestingly, the similar freezing behavior in one-norm GQD was also found in [56]. While, another new phenomenon of environment-induced double sudden changes was firstly predicted in [56] and observed in recent experiment [57]. Motivated by current studies, in this paper, we investigate the dynamics behaviors of QD [6], and one-norm GQD



Email address: [email protected]

2 [56] in two non-interacting qubits under independent classical Ornstein-Uhlenbeck (non-Markovian) noise. We consider the effect of a successive application of π-pulses on correlation dynamics. The Ornstein-Uhlenbeck noise has been discussed in various cases [41, 48, 58–62] and the method of pulse control was introduced by several authors in different systems [51, 63] to manipulate the quantum correlation. But, we have no idea about what will happen for quantum correlation in classical dephasing non-Markovian noise with pulse control. With this end, we find by studies that non-Markovian effect, initial parameters, and pulse separation time as well as pulse numbers all play an significant role in protecting QD and one-norm GQD. The freezing behavior and sudden changes of quantum correlation are studied in detail. II.

THE DYNAMICS OF TWO QUBITS WITH PULSE CONTROL

We consider the system consisting of two identical qubits, each of which is effected independently by stochastic environmental fluctuations. The Hamiltonian of the model without pulses can be written as (~ = 1) H1 = ω0 SbzA + ω0 SbzB + ΩA (t)SbzA + ΩB (t)SbzB ,

(1)

where Sbz |ei = 21 |ei , Sbz |gi = − 21 |gi, ω0 is the frequency of qubit A or B. Ωi (t) (i = A, B) are the independent classical fluctuations of the qubit level spacings, which satisfy the properties hΩi (t)i = 0, hΩi (t)Ωi (s)i =

Γi γ −γ|t−s| e , 2

(2)

where h· · ·i stands for the statistical mean over the noise ΩA (t), ΩB (t) and Γi is the dephasing damping rate of the ith qubit. The parameter γ is the noise bandwidth which determines the finite correlation time of the environmental noise by τb = γ −1 . Eq. (2) shows that the processes are non-Markovian and can reduce to the well-known Markov case when τb → 0 [64]. Essentially, a Markov process is a stochastic process with a short memory time, that is a process which rapidly forgets its past history. In such a case, the reduced state dynamics may be described using a semigroup of completely positive dynamical maps. This process has been successfully applied to the field of quantum optics. However, in realistic physical systems the assumption of Markovianity can only be an approximation, which is not justified in many situations, e.g. strong system-environment couplings and high-speed quantum information processing, etc. Then, non-Markovian process depending on past history should be taken into account in order to give right physics descriptions, in which the characteristic time scale of systems becomes comparable with the environmental correlation time. Non-Markovian process with memory has recently become of central importance in the study of open systems, many investigations have been done under non-Markovianity nowadays. In this paper, the non-Markovian noise is modeled as so-called Ornstein-Uhlenbeck process, which is Gaussian noise model. The advantage of the Ornstein-Uhlenbeck process is that the Markov limit is simply dictated by the single parameter γ. Typically, a small γ represents a non-Markovian regime and Markov limit is recovered for γ → ∞. Taking the control π-pulses into account, ones have the total Hamiltonian as H = H1 + Hp ,

(3)

where Hp is the Hamiltonian of π-pulses and given by [65] Hp = −π

X k

X 1 A −iω0 t bA iω0 t 1 B −iω0 t bB iω0 t δ(t − tk ) (Sb+ e + S− e )−π e + S− e ), δ(t − tk ) (Sb+ 2 2

(4)

k

Initially, we prepare the two qubits in three-parameter-X-type states based on the basis {|eei, |egi, |gei, |ggi} as follows   1 + c3 0 0 c1 − c2 1 0 1 − c3 c1 + c2 0  , (5) ρAB (0) =  0 c1 + c2 1 − c3 0  4 c1 − c2 0 0 1 + c3

which is called the Bell-diagonal states, i.e. the states with maximally mixed marginals. Here, ci (0 ≤ |ci | ≤ 1) are real numbers. This class of states includes the Werner states (|c1 | = |c2 | = |c3 | = c) and the Bell states (|c1 | = |c2 | = |c3 | = 1). The dynamics of the whole system can be determined by Liouville-von Neumann equation with the density operator W (t)

3

∂W (t) = −i[H, W (t)], ∂t

(6)

which in the interaction picture can be written as ∂ int W (t) = −i[HIint , W int (t)], ∂t

(7)

W int (t) = U0 (t)† W (t)U0 (t), HIint = U0 (t)† HI U0 (t), Z t dt′ H0 (t′ )]. U0 (t) = T+ exp[−i

(8)

where

0

Here T+ is the time-ordering operator and H0 (t) = ω0 SbzA + ω0 SbzB + Hp . Taking the stochastic average over the processes of Ωi (t), after a cumbersome calculation, the time-dependent reduced density matrix of two qubits with Eq. (5) in the interaction picture is given by   1 + c3 0 0 (c1 − c2 )Φ2p (t)  1 0 1 − c3 (c1 + c2 )Φ2p (t) 0 , (9) ρAB (t) =  2  0 (c1 + c2 )Φp (t) 1 − c3 0 4 2 (c1 − c2 )Φp (t) 0 0 1 + c3

Here, for n = 2m pulses, Φp (t) is

Φp (t; {δt}, k = 1, 2, ..., 2m) = exp[

2m X

k=1

gk + g(t − t2m ) + B2m ],

(10)

At the same time, for n = 2m − 1 pulses one has Φp (t; {δt}, k = 1, 2, ..., 2m − 1) = exp[

2m−1 X k=1

gk + g(t − t2m−1 ) + B2m−1 ],

(11)

where k = [t/δt] represents the largest integer not greater than t and δt is the pulse separation time. Noticed that the identical time pulses are considered in this paper. What’s more, the relevant parameters read gk = −α2 [γδt + e−γδt − 1], g(t − t2m−1 ) = −α2 [γ(t − t2m−1 ) + e−γ(t−t2m−1 ) − 1], B2m = ξ(δt)(Ξ1 − Ξ2 ) + (−1)2m ξ(t − t2m )Ξ2m , B2m−1 = ξ(δt)(Ξ1 − Ξ3 ) + (−1)2m−1 ξ(t − t2m−1 )Ξ2m−1 2m−2

2m ξ(δt) [m+r 1−r 1+r q 1−r 2 ], Ξ3 Γ e−γδt , α = 2γ .

2 1−r with ξ(y) = iα(eγy −1), Ξ1 = − ξ(δt) 1+r [(m−1)−r 1−r 2 ], Ξ2 = 2m 2m ξ(δt) r −(−1) ,Ξ2m−1 1+r

III.

=

2m−1 −(−1)2m−1 ξ(δt) r 1+r

and r =

=

(12) ξ(δt) 1−r 2m−2 1+r [m−1+r 1−r 2 ], Ξ2m

=

THE DYNAMICS OF QUANTUM CORRELATION WITH TWO QUBITS

To investigate the dynamics of quantum correlation between two noninteracting and initially correlated qubits, we present a brief review about the quantum correlation measures considered in the present study.

4 a. Quantum discord. lation, namely,

For a two-qubit system, QD is defined as the difference between total and classical correD(ρAB ) = I(ρAB ) − C(ρAB ),

(13)

where the total correlation is measured by quantum mutual information I(ρAB ) = S(ρA ) + S(ρB ) − S(ρAB ) and the classical correlation is given by C(ρAB ) = max{Bk } {S(ρA ) − S(ρAB|Bk )} with {Bk } being a set ofP projects performed locally on the subsystem B. S(ρ) = −T r(ρ log2 ρ) is the von Neumann entropy and S(ρAB|Bk ) = k pk S(ρk ) denotes (I⊗Bk ) the conditional entropy of A given the knowledge of state B, where ρk = (I⊗Bk )ρpAB is the conditional density k operator with the probability of the kth outcome of measurement pk = T r[(I ⊗ Bk )ρAB (I ⊗ Bk )]. QD is always a non-negative quantity and is zero for states with only classical correlation. Thus a nonzero value of QD indicates the presence of quantum correlation. For the X-structured density matrix in Eq. (5), following the ideal as [22], the analytic forms of classical correlation and QD can be expressed as C(ρAB ) =

2 X 1 + (−1)k χ(t) log2 [1 + (−1)k χ(t)], 2

(14)

k=1

D(ρAB ) = 2 +

4 X i=1

λi log2 λi − C(ρAB ),

(15)

where χ(t) = max{|c1 (0)|Φ2p (t), |c2 (0)|Φ2p (t), |c3 (0)|} and λi (i = 1, · · ·, 4) are the eigenvalues of Eq. (5). b. One-norm geometric quantum discord. Similar with GQD [9], the one-norm GQD is defined as [56] 1 DG (ρAB ) = min ||ρAB − ρc ||, ρc ∈Ω0

(16)

√ where ||X|| = T r[ X + X] is the one norm and Ω0 is the set of an arbitrary classical-quantum state. For the two-qubit Bell diagonal states, one-norm GQD and classical correlation can be analytically calculated as 1 1 DG (ρAB ) = int[|c1 (t)|, |c2 (t)|, |c3 (t)|], CG (ρAB ) = max[|c1 (t)|, |c2 (t)|, |c3 (t)|],

(17)

where int[· · · ] (max[· · · ]) describes the intermediate (maximal) result among the absolute values of |ci (t)|. One-norm GQD is equivalent to the negativity of quantumness [66], which is a new quantity of nonclassicality. We now begin our discussion of correlation dynamics with the measures mentioned above by examining the effect of π-pulse control and non-Markovian properties on them. For simplicity, in this paper, we only consider the initial parameters as c1 = 1, c2 6= c3 and the same dephasing damping rate Γ1 = Γ2 = Γ is taken. Figs. 1(a) and 1(b) show the QD dynamics with weak non-Markovian effect b = 3 and the initial parameters c2 = −c3 = −0.8. While, Figs. 1(c) and 1(d) are the case of strong non-Markovian effect b = 0.6 with the same initial parameters as given by Figs. 1(a) and 1(b). Here, we set the parameter b = γ/Γ, which reflects the strength of non-Markovian noise and controls the approach to the Markov limit. Typically, a small b represents a non-Markovian regime. We can see that both the non-Markovian properties and time difference Γδt between adjacent pulses play an important role in shielding the correlation. The stronger non-Markovian effect is, the longer time of sudden transition tc of QD will happen, which is consistent with our previous depictions [48]. Although the dynamics decoupling pulses are always helpful for improving QD, some constraints about the pulses are set to maintain the phenomenon of sudden transition of QD reported in [47]. It could be clearly seen from dashed line (14 pulse numbers are run) and dotted line (12 pulse numbers) in Fig. 1(a) that the behavior of ”freezing-collapse-freezing-oscillating decay”(FCFOD) happens for QD when the pulses with long time interval (after we call them as long time pulses) Γδt = 0.5 is applied. With the increase of pulse numbers, QD will be improved further accompanying the behavior of oscillating decay. However, the freezing time of QD cannot be prolonged anymore. In order to obtain longer sudden transition time tc of QD, we can fall back on short-time-difference pulses (or short time pulses), as shown in Fig. 1(b) in which 30 pulse numbers are applied for solid line, 40 and 44, respectively, for dotted and dashed lines and Γδt = 0.2. It is quite clear from the solid line in Fig. 1(b) that tc can be prolonged to a great extent. But tc cannot be lengthened at will since QD will display FCFOD behavior if the pulse numbers are greater than a certain critical value, which can be seen from dotted and dashed lines in Fig. 1(b). Of course, if shorter time pulses are used, longer tc we can obtain. In addition, one can find from Figs. 1(c) and 1(d) that strong non-Markovian properties also are crucial for QD dynamics. Besides the similar results are got from Fig. 1(c) and Fig. 1(d) as previous both of figures. A significant difference between them is that long time pulses are also advantageous for extending tc . Here 14 pulse numbers are applied for dashed

5 line and 12 for dotted one in Fig. 1(c) with Γδt = 1. And, in Fig. 1(d), 50, 59 and 63 pulse numbers, respectively, for solid, dotted as well as dashed lines with Γδt = 0.5. Interestingly, [56] found that one-norm GQD can also exhibit freezing behavior during its evolution. In particular, the new phenomenon of double sudden changes of one-norm GQD was reported firstly in [56] and confirmed in recent experiment [57]. Fig. 2 compares one-norm GQD with QD using the same initial parameters as Fig. 1. One can see that one-norm GQD is always greater than QD and the sudden transition time tc of one-norm GQD is longer slightly than QD. This result also comes into existence in strong non-Markovian case. When the same π-pulses are used as Fig. 1, GQD displays similar FCFOD behavior as QD, however, it owns more smaller oscillating amplitude and longer freezing time than QD. So, the related depictions on GQD dynamics under pulse control can see QD of Fig. 1. In the following, we discuss the behavior of double sudden changes on one-norm GQD. In Fig. 3, we plot the dynamics of one-norm GQD as a function of the dimensionless scaled time Γt with b = 3 in (a), b = 0.5 in (b) and c2 = −0.8, c3 = 0.7 for dashed line, c2 = −0.8, c3 = 0.2 for solid line. It is shown that the maximal value and the first sudden transition point (FSTP) of one-norm GQD are determined by |c2 | and |c3 |, respectively. When given c2 , FSTP of one-norm GQD will occur more later with the decrease of c3 , and the corresponding freezing time of one-norm GQD is shorter than greater c3 . However, ultimately, the change trend of one-norm GQD will be agreement after double sudden changes for all c3 . Moreover, we know from Fig. 3 that strong non-Markovian effect will prolong the freezing time of one-norm GQD and postpone the occurrence of FSTP. As supplement, in Fig. 4, we show the change of one-norm GQD with different parameters c2 = −0.3, c3 = 0.2. It can be seen that smaller c2 will increase further the freezing time of one-norm GQD and speed up the emergence of FSTP. The influence of π-pulses on double sudden changes (DSC) is illustrated in Figs. 5-9 with c2 = −0.8. It is shown that short Γδt is more beneficial than long for enlarging the freezing time of DSC in strong Markovian regime, which can be seen from Figs. 5-7 with b = 3. However, there is a range of c3 when control pulses are exerted to keep the behavior of DSC. In Fig. 5, only one pulse with Γδt = 1 is used since DSC will disappear for more than one pulse. In this case, when c3 ∈ [0.444, 0.505) there exists DSC (see as solid line) in Fig. 5. With Γδt becomes smaller, the upper and lower bounds of c3 will rise up overall. The lower bound in Fig. 6 is 0.675 in which three pulses are used and the upper bound corresponds to 0.785 with one pulse for Γδt = 0.5. At the same time, there is DSC in Fig. 7 for c3 ∈ [0.781, 0.8) with Γδt = 0.2. It is not difficult to deduce that for smaller Γδt than 0.2, the lower bound of c3 shall gradually move to 0.8 until DSC becomes single sudden transition. Striking contrast to weak non-Markovian case, we find DSC and the freezing time can be remained better by the aid of longer Γδt with the increase of non-Markovian effect, which can be demonstrated from Figs. 8 and 9. In Fig. 8, we choose c3 = 0.787, 25 control pulses in (a) and c3 = 0.797, one pulse in (b), other parameters read c2 = −0.8, b = 0.6, Γδt = 0.5. We can see that DSC happens in (a); however, DSC almost is single sudden transition in (b). It’s worth noting that DSC has vanished for Γδt = 0.2 control pulses and the relevant figures of one-norm GQD are neglected for the sake of brevity. Fig. 9 is the case of DSC with 24 control pulses for Γδt = 2, b = 0.1, c2 = −0.8 and c3 = 0.769. Here and now, the pulses with Γδ < 1.0 can not retain DSC any more. Another point we wolud like to mention is that the lower bound of c3 goes down with the increasing of Γδ in strong non-Markovian properties, which can be understood by comparing Fig. 8 with Fig. 9. IV.

CONCLUSION

In this paper, we have investigated the dynamics of quantum correlation for two noninteracting qubits, each locally interacting with its own classical dephasing non-Markovian environment. By preparing the three-parameter-X-type initial states, the influence of non-Markovian effect and dynamics decoupling π-pulses on quantum discord and onenorm geometric quantum discord was demonstrated. It was shown that depending on non-Markovian properties, the freezing behavior of single sudden change in quantum correlation can be kept for apt pulse separation time and pulse numbers. However, if these conditions are violated then the quantum correlation will present complex behaviors, for example, oscillating decay, etc. In addition, it was shown that the amplitude (freezing time) of one-norm geometric quantum discord is always greater (a bit longer) than quantum discord. Finally, we have discussed the double sudden changes of one-norm geometric quantum discord by means of π-pulses. We have found that short time pulses are more suit to enhance the freezing time of double sudden changes in strong Markovian regime while long time pulses for strong non-Markovian one. However, in order to achieve the behavior of double sudden changes when pulses are applied, the initial parameters are bounded. V.

ACKNOWLEDGMENTS

This work was supported by the Natural Science Foundation of China under Grants Nos. 11105087, 61275210, 11275118, 11404198,11302121 and the Youth Science Foundation of Shanxi Province of China under Grant No.

6 2010021003-2.

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Figure captions: Fig. 1: Time evolution of QD D(ρ(t)) described by the initial parameters c1 = 1, c2 = −c3 = −0.8 as a function of the dimensionless time Γt with b = 3 in (a) and (b) and b = 0.6 in (c) and (d). (a) solid line is the case without control pulses. The dashed and dotted lines correspond to 14 and 12 control pulse numbers with Γδt = 0.5 (Γδt is the identical separation time between pulses), respectively. (b) 30, 40 and 44 control pulses are used in solid, dotted and dashed lines, respectively with Γδt = 0.2. (c) dashed line is 14 pulses numbers while 12 for dotted one with Γδt = 1. (d) 50, 59 and 63 pulse numbers, respectively, for solid, dotted as well as dashed lines with Γδt = 0.5. Here, we set the parameter b = γ/Γ, which reflects the strength of non-Markovian noise and controls the approach to the Markov limit. 1 Fig. 2: QD D(ρ(t)) and one-norm GQD DG (ρ(t)) as a function of dimensionless scaled time Γδt with b = 1 and the same initial parameters as Fig. 1. 1 Fig. 3: The dynamics of one-norm GQD DG (ρ(t)) as a function of the dimensionless scaled time Γt with b = 3 in (a), b = 0.5 in (b) and c1 = 1, c2 = −0.8, c3 = 0.7 for dashed line, c1 = 1, c2 = −0.8, c3 = 0.2 for solid line. Fig. 4: One-norm GQD versus Γt with c1 = 1, c2 = −0.3, c3 = 0.2. 1 Fig. 5: Solid line denotes the plot of DG (ρ(t)) versus Γt, dotted line describes the change of classical correlation 1 CG (ρt ). Meanwhile, dashed line is the minimum value among |c1 (t)|, |c2 (t)| and |c3 (t)|. The relevant parameters in (a) is c1 = 1, c2 = −0.8, c3 = 0.444, b = 3 and c1 = 1, c2 = −0.8, c3 = 0.505, b = 3 in (b). Thereinto, only one Γδt = 1 pulse is used in (a) and (b). Fig. 6: Solid, dotted and dashed lines have the same meanings as Fig. 5. (a) 3 pulse numbers are used and c3 = 0.675, Γδt = 0.5; (b) one pulse number is applied and c3 = 0.785, Γδt = 0.5. The other parameters see Fig. 5. Fig. 7: Solid, dotted and dashed lines have the same meanings as Fig. 5. The other parameters are c1 = 1, c2 = −0.8, c3 = 0.781, Γδt = 0.2 and b = 3. Here 34 pulse numbers are used. Fig. 8: (a) c1 = 1, c2 = −0.8, c3 = 0.787, b = 0.6, Γδt = 0.5 and 25 pulses; (b) c3 = 0.797, one pulse and other parameters are the same as (a). Fig. 9: One-norm geometric quantum discord versus Γt with 24 control pulses, Γδt = 2, b = 0.1, c2 = −0.8, c3 = 0.769.

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