PHYSICAL REVIEW A 76, 032322 共2007兲
Controllable dynamics of two separate qubits in Bell states 1
Jun Jing,1,* Zhi-guo Lü,2,† and Guo-hong Yang1
Department of Physics, Shanghai University, Shanghai 200444, China Department of Physics, Shanghai Jiaotong University, Shanghai 200240, China 共Received 23 May 2007; published 21 September 2007兲
2
The dynamics of entanglement and fidelity for a subsystem of two separate spin-1 / 2 qubits prepared in Bell states is investigated. One of the subsystem qubit labeled A is under the influence of a Heisenberg XY spin bath, while another one labeled B is uncoupled with that. We discuss two cases: 共i兲 the number of bath spins is infinite, N → ⬁, and 共ii兲 N is finite, N = 40. In both cases, the bath is initially prepared in a thermal equilibrium state. It is shown that the time dependence of the concurrence and the fidelity of the two subsystem qubits can be controlled by tuning the parameters of the spin bath, such as the anisotropic parameter, the temperature, and the coupling strength with qubit A. It is interesting to find that the dynamics of the concurrence is independent of four different initial Bell states and that of the fidelity is divided into two groups. DOI: 10.1103/PhysRevA.76.032322
PACS number共s兲: 03.67.Mn, 75.10.Jm
I. INTRODUCTION
Entanglement, which exhibits a very peculiar correlation among the degrees of freedom of a single particle or the distinct parts of a composite system, is the most intriguing feature of quantum composite systems and a vital resource for quantum computation and quantum communication 关1–6兴. In the field of quantum information theory, it is a fundamental issue to create, quantify, control, and manipulate the entangled quantum bits, which are often composed of spin-1 / 2 atoms in different problems 关1,7–9兴. Particularly, many works 关10–12兴 are devoted to steering two initially entangled qubits through an auxiliary particle or field 共for instance, another spin qubit or a bosonic mode兲, which interacts with only one of them. It is a very exciting motivation, yet those approximated models neglect the actual complex environment of the quantum qubits. And it remains an important open question how the entanglement degree responds to the influence of environmental noise 关13兴. Practically, spin qubits are indeed open quantum subsystems and exposed to the influence of their environments 关14–17兴. In most conditions, the coupling between the subsystem and environment will degrade the entanglement degree between the subsystem qubits. In some other conditions, however, a specially structured and well-designed bath can be conceived as a protection device to suppress the negative influence from itself or other noise sources 关18–20兴. For the spin subsystem, there are two important modes of baths: 共i兲 boson bath—e.g., the Caldeira-Leggett model 关21兴; 共ii兲 spin bath—e.g., the model used in Ref. 关22兴. Here we discuss the latter one. It is well known that the localized spins in solid-state nanodevices, the most promising candidates for qubits due to their easy scalability and controllability 关7,23兴, are mainly subject to the influence from the nuclear spins, which constitute a type of spin-1 / 2 environment. It is an almost intractable computation task to deal with such a spin-spin-bath model because of its giant number
*
[email protected] †
[email protected]
1050-2947/2007/76共3兲/032322共8兲
of degrees of freedom. Therefore, physicists resort to some approximations or simplifications, such as Markovian 关24兴 schemes and the non-Markovian ones 关25兴, which have been developed in the past two decades. Based on these schemes, plenty of analytical and numerical methods have been exploited to study the reduced dynamics of subsystems consisting of spins-1 / 2 by tracing out the degrees of freedom of the spin bath. Some recent works focused on the center spins in a network configuration, in which the form of bath is specially structured, such as a thermal bath 关26,27兴, a bath via Heisenberg XX couplings 关28兴, and a thermal spin bath via Heisenberg XY couplings 关17,29兴. In this paper, an open two-spin-qubit subsystem 共two qubits labeled A and B, respectively兲 is explored as a target quantum information device with a spin bath of a starlike configuration. The model is something like the one considered in Refs. 关17,30兴. But there are significant differences between them. It is supposed that at the beginning, the subsystem is prepared as one of the Bell states 共EinsteinPodolsky-Rosen pairs兲 关31,32兴 兩e1典 = 1/冑2共兩11典 + 兩00典兲, 兩e2典 = 1/冑2共兩10典 + 兩01典兲, 兩e3典 = 1/冑2共兩11典 − 兩00典兲, 兩e4典 = 1/冑2共兩10典 − 兩01典兲. Then qubit B is moved far away or isolated so that not only the coupling between qubits B and A, but also the interaction of B with the spin bath could be neglected. The bath is regarded as an adjustable auxiliary device to control the time evolution of the two qubits. And the evolution is represented by the concurrence and fidelity dynamics as functions of various parameters associated with the bath. The reduced dynamics of the subsystem is obtained by a numerical scheme combined with the Holstein-Primakoff transformation and the Laguerre polynomial expansion algorithm. We consider two conditions in which the number of spins in the bath is infinite and finite. The rest of this paper is organized
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©2007 The American Physical Society
PHYSICAL REVIEW A 76, 032322 共2007兲
JING, LÜ, AND YANG
as follows. In Sec. II we first give the model Hamiltonian and its analytical derivation, and then we introduce the numerical calculation of the evolution of the reduced matrix for the subsystem. Detailed results and a discussion are given in Sec. III. The conclusion is given in Sec. IV. II. MODEL AND METHOD
The subsystem consists of two entangled spin-1 / 2 atoms labeled A and B respectively, between which there is no coupling. Qubit A interacts with a spin-1 / 2 bath via a Heisenberg XY interaction while B does not. The total system Hamiltonian, similar to those considered in Refs. 关16,17,28,33兴, is divided into Hs, Hb, and Hsb. They represent the subsystem, the bath, and the interaction 关16,17,34兴 parts between the former two terms, respectively: Hs = Hsb =
+
Bz兲,
兺 关共1 +
2冑N i=1
␥兲Axxi
with 关b , b 兴 = 1 and the first-order approximation of 1 / N,
冑
Hsb = g0
␥兲Ayiy兴,
冋冉
b† −
where 0 is half of the energy bias for the two-level atom A or B, g0 is the coupling strength between subsystem spin A and the bath spins, and g represents the mutual interactions among the bath spins. ␥ 共0 艋 ␥ 艋 1兲 is the anisotropic parameter. When ␥ = 0, the XY interaction is reduced to an XX one 关17兴. The x, y, and z components of the matrix are the Pauli matrices
冉 冊 0 1 1 0
,
y =
冉 冊 0 −i i
0
z =
,
冉 冊 1
0
0 −1
.
共4兲
The indices i in the summation run from 1 to N, and N is number of bath spins. Adopting ± = 1 / 2共x ± iy兲, we can rewrite the Hamiltonians 共2兲 and 共3兲 by Hsb =
g0
冑N
冋
N
N
i=1
i=1
册
兺 +i 共␥A+ + A−兲 + 兺 −i 共A+ + ␥A−兲 ,
共5兲
N
Hb =
g 兺 关␥共+i +j + −i −j 兲 + 共+i −j + −i +j 兲兴. N i⫽j
Hsb =
g0
冑N 关J+共␥A + A兲 + J−共A + ␥A兲兴, +
−
+
−
册
冊
b†nˆ 2N
b† −
b−
nˆ − 1 2N
b†nˆ nˆb + b− 2N 2N
nˆb nˆb + b− 2N 2N
nˆ + 1 2N
+ 2nˆ +
1−
b† −
b−
nˆb 2N
b†nˆ −1 2N
nˆ − 2 nˆ + b2 1 − 2N 2N
nˆ共2nˆ2 − 8Nnˆ − nˆ + 1兲 . 4N2
共11兲
Utilizing the collective environment pseudospin J and the Holstein-Primakoff transformation, one could reduce a highsymmetric spin bath, such as the one we considered, to a single-mode bosonic bath field 关16,17兴. The transformed Hamiltonian is just like a spin-boson model in the field of cavity quantum electrodynamics 共CQED兲. And the effect of the single-mode bath on the dynamics of the two subsystem qubits is interesting although the bath only directly interacts with one of them. The model might be helpful to understand the magic essence of quantum entanglement and practical in manipulating the quantum communication. B. Calculation method
The whole state of the total system is assumed to be separable before t = 0—i.e., 共0兲 = 兩共0兲典具共0兲 兩 丢 b. The subsystem 兩共0兲典 is prepared as one of the four Bell states 兩ei典, i = 1 , 2 , 3 , 4. The bath is in a thermal equilibrium state b共0兲 = e−Hb/kBT / Z, where Z = Tr共e−Hb/kBT兲 is the partition function. The Boltzmann constant kB is set to 1 for the sake of simplicity in later calculations. To derive the density matrix 共t兲 of the whole system,
共6兲
Substituting the collective angular momentum operators J± N = 兺i=1 ±i 关35兴 into Eqs. 共5兲 and 共6兲, we get
冉
b†nˆ 2N
= g ␥ 共b†兲2 1 −
N
共3兲
b† −
+ b† −
共2兲
g Hb = 兺 关共1 + ␥兲xi xj + 共1 − ␥兲iyyj 兴, 2N i⫽j
冊
b†nˆ nˆb 共␥A+ + A− 兲 + b − 共A+ + ␥A− 兲 , 2N 2N
再 冋冉 冊冉 冊 冉 冊冉 冊册 冉 冊冉 冊 冉 冊冉 冊 冎 再 冋 冉 冊冉 冊 冉 冊 冉 冊册 冎
Hb = g ␥
⫻ 1−
x =
N − b †b b †b nˆ =1− =1− , N 2N 2N
共10兲
共1兲
+ 共1 −
共9兲
†
N
g0
J− = 共冑N − b†b兲b,
the Hamiltonians 共7兲 and 共8兲 can be written as
A. Hamiltonian
0共Az
J+ = b†共冑N − b†b兲,
共t兲 = exp共− iHt兲共0兲exp共iHt兲,
共12兲
we need to consider two factors. 共i兲 To express the thermal bath, we use the method suggested by Tessieri and Wilkie 关18,20,29兴:
共7兲
N
b共0兲 =
兺 兩m典m具m兩,
共13兲
m=1
Hb =
g 关␥共J+J+ + J−J−兲 + 共J+J− + J−J+ − N兲兴. N
共8兲
m =
By the Holstein-Primakoff transformation 关36兴, 032322-2
e−Em/T , Z
共14兲
PHYSICAL REVIEW A 76, 032322 共2007兲
CONTROLLABLE DYNAMICS OF TWO SEPARATE QUBITS… N
Z=
兺 e−E /T , m=1 m
III. SIMULATION RESULTS AND DISCUSSIONS
共15兲
where 兩m典, m = 1 , 2 , 3 , . . . , N, are the eigenstates of the environment Hamiltonian Hb and Em the corresponding eigenenergies in increasing order. On the condition of the thermodynamics limit—i.e. N → ⬁—Eqs. 共10兲 and 共11兲 are simplified as Hsb = g0关b†共␥A+ + A− + ␥B+ + B− 兲 + b共A+ + ␥A− + B+ + ␥B− 兲兴, Hb = g关␥共b†2 + b2兲 + 2b†b兴.
共16兲 共17兲
Then N in Eqs. 共13兲 and 共15兲 should be replaced with a cutoff M lined to a certain high-energy level. By the above expansion, the initial state can be represented by
共0兲 = 兺 m兩⌿m共0兲典具⌿m共0兲兩,
兩⌿m共0兲典 = 兩共0兲典兩m典. 共18兲
共ii兲 For the evaluation of the evolution operator U共t兲 = exp共iHt兲, we apply the Laguerre polynomial expansion scheme, which is proposed by us 关20,29,37兴, into the computation:
冉 冊 兺冉 冊
1 U共t兲 = 1 + it
␣+1 ⬁
k=0
it k ␣ Lk 共H兲. 1 + it
共19兲
Lk␣共H兲 is one type of Laguerre polynomials 关38兴 as a function of H, where ␣ 共−1 ⬍ ␣ ⬍ ⬁兲 distinguishes different types of Laguerre polynomials and k is the order of them. In real calculations the expansion has to be cut at some value of kmax, which was optimized to be 20 in this study 共we have to test out a kmax for a compromise of the numerical stability in the recurrence of the Laguerre polynomial and the speed of calculation兲. With the largest order of the expansion fixed, the time step t is restricted to some value in order to get accurate results of the evolution operator. At every time step, the accuracy of the results will be confirmed by the test of numerical stability—whether the trace of the density matrix is 1 with error less than 10−12. For longer time, the evolution can be achieved by more steps. The action of the Laguerre polynomial of the Hamiltonian to the states is calculated by recurrence relations of the Laguerre polynomial. The scheme is of an efficient numerical algorithm motivated by Refs. 关39,40兴 and is pretty well suited to many quantum problems, open or closed. It could give results in a much shorter time compared with the traditional methods, such as the wellknown fourth-order Runge-Kutta algorithm, under the same requirement of numerical accuracy. After some derivations, the density matrix of the whole system 共t兲 can be determined by Eqs. 共12兲 and 共18兲. Tracing out the degrees of freedom of the environment, we finally obtain the dynamics of the subsystem qubits:
s共t兲 = Trb关共t兲兴.
共20兲
With s共t兲, we can discuss 共i兲 the concurrence 关41,42兴, which is a very good measurement for the intraentanglement of two two-level particles and defined as C = max兵1 − 2 − 3 − 4,0其,
共21兲
where i, i = 1 , 2 , 3 , 4, are the square roots of the eigenvalues of the product matrix s共y 丢 y兲s*共y 丢 y兲 in decreasing order; 共ii兲 the fidelity 关43兴, which is defined as Fd共t兲 = Tr关ideal共t兲s共t兲兴.
共22兲
ideal共t兲 represents the pure-state evolution of the subsystem under Hs only, without interaction with the environment. The fidelity is a measure for decoherence and depends on ideal. It achieves its maximum value 1 only if the time-dependent density matrix s共t兲 is equal to ideal共t兲. The results and a discussion of the two quantities are divided into two subsections, Secs. III A and III B. Both of these two cases suggest a type of controlling method to the entangled qubits, especially the latter one. It is interesting to note that no matter which one of the four Bell states is chosen as the initial state for the subsystem, there is no distinction between their dynamics of concurrence. So we need not to concretely point out the initial states in the following discussions. But there are differences between the fidelity dynamics of 兩e1,3典 and 兩e2,4典. It is shown that the spins in the initial state of the first group, 兩e1典 and 兩e3典, are parallel, while in those of the second group, 兩e2典 and 兩e4典, are antiparallel. That is why the two groups have different fidelity evolution behaviors. In fact, any other basis obtained from the Bell states by replacing the coefficients ±1 / 冑2 with ei / 冑2 共 being a real number兲 has the same dynamics 关44兴. A. Thermodynamic limit „N \ ⴥ…
The free evolution 共it means the subsystem is decoupled from the bath or g0 = 0兲 of both concurrence and fidelity of the subsystem is time independent, because 共i兲 Hs in Eq. 共1兲 关Hs = 0共Az + Bz兲兴 cannot change the entanglement degree between the two qubits; 共ii兲 when g0 = 0, the effect of the bath is excluded from the evolution of the subsystem. Thus, if g0 = 0, we always have C共t兲 = Fd共t兲 = 1. In Fig. 1共a兲, we show different effects on the dynamics by four anisotropic parameters: ␥ = 0 , 0.2, 0.6, 1. From the four curves, we obtain two findings. 共i兲 When ␥ is not too large, the entanglement 关C共0兲 = 1兴 can always be recovered to a high degree after some oscillations. For instance, C共␥ = 0 , g0t = 8.960兲 = 0.974 631, C共␥ = 0.2, g0t = 8.912兲 = 0.961 513, and C共␥ = 0.6, g0t = 12.672兲 = 0.844 158. Yet the concurrence will never reach 1 in a long time scale. 共ii兲 The curve of ␥ = 1 shows a totally different behavior from the other three cases. It keeps decreasing with some small fluctuations. Then we turn to the other subfigures. Figure 1共c兲 is almost the same as Fig. 1共b兲. There are some obvious disagreements between Figs. 1共b兲 and 1共a兲. For example, C共␥ = 0 , g0t = 8.960兲 = 0.974 631, but at the same time, Fd1共␥ = 0 , g0t = 8.960兲 = 0.101 803. In other words, when the concurrence of the
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PHYSICAL REVIEW A 76, 032322 共2007兲
JING, LÜ, AND YANG
1
0.8
0.8
0.6
0.6
C(t)
C(t)
1
0.4
0.4
0.2
0.2
0
0
0
5
10
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1 10
15
0 0
20
g0t
0.8
0.8
0.7
0.7
0.6
0.6
Fd(t)
1 0.9
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1 15
0 0
20
g0t
(c)
20
10
15
20
15
20
0.5
0.4
10
15
g0t
1
5
5
(b)
0.9
0 0
10
0.5
0.4
5
5
g0t 1
Fd(t)
Fd(t)
0
(a)
1
(b)
Fd(t)
20
0
0 0
(c)
15
g t
(a)
5
10
g t 0
FIG. 1. Time evolution of 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different anisotropic parameters: ␥ = 0 共solid curve兲, ␥ = 0.2 共dashed curve兲, ␥ = 0.6 共dot-dashed curve兲, and ␥ = 1.0 共dotted curve兲. Other parameters are N → ⬁, 0 = 2g0, g = g0, and T = g0.
FIG. 2. Time evolution for 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different values of temperature: T = 0.2g0 共solid curve兲, T = g0 共dot-dashed curve兲, and T = 5g0 共dashed curve兲. Other parameters are N → ⬁, 0 = 2g0, g = g0, and ␥ = 0.6.
subsystem has been mostly retrieved, the state of that is not simultaneously back to its initial state. Only the combination of the concurrence and the fidelity can give a complete description of the real revival of the state. It can be testified in the case of ␥ = 0 共solid curves in the two subfigures兲. When
g0t = 15.624, the concurrence evolves to C = 0.875 742 and the fidelity goes back to Fd1 = 0.924 277. So at that moment, s共g0t兲 is mainly composed by 兩e1,3典具e1,3兩. In Figs. 2共a兲–2共c兲, we plot the dynamics of the concurrence and fidelity at different temperatures. When the tem-
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CONTROLLABLE DYNAMICS OF TWO SEPARATE QUBITS…
perature is not too high, such as T = 0.2g0 and T = 1g0, both concurrence and fidelity represent a periodical oscillation. At some moments, they can be restored to a high degree. The restoring degree of both quantities, however, decreases as the temperature increases. Similar to Fig. 1, revivals of the concurrence and fidelity do not take place simultaneously. When the bath is at a high temperature, such as T = 5g0, the concurrence quickly declines to zero 关see the dashed line in Fig. 2共a兲兴 and does not go back to C ⬎ 0 immediately. It means that when the local spin bath is adjusted to a high temperature, it makes a sudden disappearance to the entanglement of a nonlocalized state and it will lose the control ability of the subsystem. This is the effect that has been called “entanglement sudden death” 共ESD兲 关45,46兴. In Ref. 关46兴, after the concurrence goes abruptly to zero, it arises more or less from nowhere, since there is no local effect under the action of weak noises. Our model is still an example of ESD; however, the concurrence arises after some time due to the local thermal bath. To find out the role of the subsystem-bath coupling strength g, we keep the bath at a moderate temperature T = g0. In Fig. 3共a兲, all three curves show periodical behaviors and the oscillation amplitudes are strikingly damped by increasing g from 2g0 to 8g0. When g increases to 8g0, the fluctuation magnitude of concurrence near C = 1 is too small to be noticed. It is like the case of g0 = 0, in which the bath is decoupled from the subsystem and C共t兲 = 1. It is consistent with the claims in Refs. 关18–20兴 that enough strong intracoupling strength among bath spins can make the evolution of the subsystem be completely determined by the Hamiltonian of itself Hs. But the subsystem state does not receive the same protection as the subsystem entanglement degree, especially when g = 8g0. Figure 3共b兲 manifests that the revival period of fidelity is much longer than that of concurrence. In fact, because qubit B is not under the influence of the bath, the bath cannot make a decoherence-suppression effect on the subsystem.
1
0.8
C(t)
0.6
0.4
0.2
0 0
5
10
15
20
15
20
15
20
g t
(a)
0
1 0.9 0.8 0.7
Fd(t)
0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
g0t
(b) 1 0.9 0.8 0.7
Fd(t)
0.6
B. Finite bath spins „N = 40…
0.5 0.4
In the previous two-center–spin-spin-bath works 关17,29兴, it is supposed that the number of bath spins is infinite, which helps to reduce the Hamiltonian 共10兲 and 共11兲 to a simple form. Yet as the controlling device in real quantum information equipment, the spin bath, in principle, should be made of a finite number of spins 1 / 2. Then, in this subsection, we use the first-order expansion of the Hamiltonians 共10兲 and 共11兲 to introduce a finite N to the present problem. The error of this approximation is about O共1 / N2兲 as Eq. 共11兲 indicates. Without loss of generality, we set N = 40. Comparing the result of Fig. 1 with that of Fig. 4, we can find some agreements and some disagreements. The most identical characteristic between them is that the concurrence dynamics is independent of the choice of state as long as it is one of the four Bell states. Yet when ␥ = 1, the concurrence 共dotted curve兲 does not decrease monotonously during the given time. For fidelity, it is shown that with a larger anisotropic parameter ␥, the curves in Fig. 4共b兲 oscillate with a shorter period, which is opposite to the tendency in Fig. 1共b兲,
0.3 0.2 0.1 0 0
(c)
5
10
g t 0
FIG. 3. Time evolution for 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different values of coupling strength between subsystem and bath: g = 2g0 共solid curve兲, g = 4g0 共dashed curve兲, and g = 8g0 共dot-dashed curve兲. Other parameters are N → ⬁, 0 = 2g0, ␥ = 0.6, and T = g0.
while Fig. 4共c兲 is almost the same as Fig. 1共c兲, with a little longer oscillation period. It seems that the fidelity evolution of the second group is not very sensitive to the bath-spin number N. The differences between the infinite-N and finiteN cases might arise from their different energy-level num-
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PHYSICAL REVIEW A 76, 032322 共2007兲
JING, LÜ, AND YANG
1
0.8
0.8
0.6
0.6
C(t)
C(t)
1
0.4
0.4
0.2
0.2
0
0
0
5
10
0.9 0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1 15
0 0
20
g0t
0.8
0.8
0.7
0.7
0.6
0.6
Fd(t)
1 0.9
0.4
0.3
0.3
0.2
0.2
0.1
0.1 15
0 0
20
g t
(c)
0
15
20
15
20
0.5
0.4
10
10
0
1
5
20
g t
0.9
0 0
5
(b)
0.5
15
0.5
0.4
10
10
0
0.8
5
5
g t
0.9
Fd(t)
Fd(t)
0
(a) 1
(b)
Fd(t)
20
1
0 0
(c)
15
g0t
(a)
5
10
g0t
FIG. 4. Time evolution for 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different values of anisotropic parameter: ␥ = 0 共solid curve兲, ␥ = 0.2 共dashed curve兲, ␥ = 0.6 共dot-dashed curve兲, and ␥ = 1.0 共dotted curve兲. Other parameters are N = 40, 0 = 2g0, g = g0, and T = g0.
FIG. 5. Time evolution for 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different values of temperature: T = 0.2g0 共solid curve兲, T = g0 共dot-dashed curve兲, and T = 5g0 共dashed curve兲. Other parameters are N = 40, 0 = 2g0, g = g0, and ␥ = 0.6.
bers and corresponding weights in our numerical scheme 共in Sec. III B兲. Under the same requirement of numerical accuracy, for an infinite N, we need to consider 14, 15, 18, and 20 energy levels when ␥ is 0, 0.2, 0.6, and 1, respectively; for N = 40, we calculate 9, 10, 17, and 18 levels, respectively.
In a comparison of Fig. 5 with Fig. 2, we can also find the effect of a finite N. At low temperature 共T = 0.2g0兲, the entanglement degree of the subsystem qubits oscillates with a nearly perfect period between the value of 0.8 and 1.0 关solid curve in Fig. 5共a兲兴. However, the subsystem of group 1
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destroy this perfect oscillation. When T = 1g0, the second peak value C共g0t = 16.19兲 = 0.918 798 is lower than the first one C共g0t = 0兲 = 1 关see the dot-dashed curve in Fig. 5共a兲兴 and the peak Fd1共g0t = 11.55兲 = 0.877 301 is higher than Fd1共g0t = 23.13兲 = 0.779 953 关see the dot-dashed curve in Fig. 5共b兲兴. When the temperature is up to 5g0, the entanglement vanishes to zero in a fairly short stretch of time. The entanglement “death” time is longer than that in the case of N → ⬁ 共Fig. 2兲. So whether N → ⬁ or N is finite, the revival of the concurrence after ESD results from the effect of thermal bath. For the subsystem-bath coupling g, the dynamics of concurrence in the case of N = 40 关see Fig. 6共a兲兴 is almost the same as that in the N → ⬁ case 关compare with Fig. 3共a兲兴. The evolution of the fidelity of the first group shows significant changes when N is changed from infinity 关Fig. 3共b兲兴 to 40 关Fig. 6共b兲兴 while the fidelity of the second group does not show very obvious changes. And for the first group, all the three cases behave periodical oscillations. They manifest that the subsystem in the condition of N = 40 can be restored to the initial state with more chances or possibilities than that in the condition of N → ⬁.
1
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IV. CONCLUSION
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FIG. 6. Time evolution for 共a兲 concurrence from all four Bell states, 共b兲 fidelity from 兩e1,3典, and 共c兲 fidelity from 兩e2,4典 at different values of coupling strength between subsystem and bath: g = 2g0 共solid curve兲, g = 4g0 共dashed curve兲, and g = 8g0 共dot-dashed curve兲. Other parameters are N = 40, 0 = 2g0, ␥ = 0.6, and T = g0.
关group 2兴 goes back to its own initial state only once in almost five 关ten兴 revival periods of concurrence, which is illustrated by the corresponding curve in Fig. 5共b兲 关Fig. 5共c兲兴. It is obvious that the increase of the temperature will also
We studied the time evolution of two separated qubit spins with a thermal equilibrium bath composed of infinite or finite spins in a quantum anisotropic Heisenberg XY model. The bath can be treated effectively as a single pseudospin of N / 2 according to the symmetry of the Hamiltonian. By the Holstein-Primakoff transformation and the first order of the 1 / N expansion, it is further considered as a single-mode boson field. The pair of qubits serving as an quantum information device is initially prepared in a Bell state. It is interesting that the concurrence and the fidelity dynamics of the subsystem can be controlled by some characteristic parameters of the spin bath. Through the adjustment, we show that 共i兲 the concurrence dynamics of the subsystem is independent of the initial state, whether N is infinite or finite; however, the fidelity dynamics is divided into two groups; 共ii兲 a smaller anisotropic parameter ␥ can help the subsystem to evolve into a highly entangled state, but this restoration should be measured by the combination of concurrence and fidelity; 共iii兲 the bath at higher temperature makes a sudden death to the entanglement 共ESD兲 of the subsystem and strongly destroys the fidelity of that; 共iv兲 the spin bath can help to keep the high entanglement degree between the two subsystem spins in the condition of large intracoupling g. ACKNOWLEDGMENTS
We would like to acknowledge the support from the National Natural Science Foundation of China under Grant No. 10575068, the Natural Science Foundation of Shanghai Municipal Science Technology Commission under Grants Nos. 04ZR14059 and 04dz05905, and the CAS Knowledge Innovation Project No. KJcx.syw.N2.
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JING, LÜ, AND YANG 关1兴 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information 共Cambridge University Press, Cambridge, England, 2000兲. 关2兴 C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 共1996兲. 关3兴 R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, e-print arXiv:quant-ph/0702225. 关4兴 R. F. Werner, Phys. Rev. A 40, 4277 共1989兲. 关5兴 D. M. Greenberger, M. Horne, and A. Zeilinger, Bells Theorem, Quantum Theory, and Conceptions of the Universe 共Kluwer Academic, Dordrecht, 1989兲. 关6兴 W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 共2000兲. 关7兴 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 共1998兲. 关8兴 B. E. Kane, Nature 共London兲 393, 133 共1998兲. 关9兴 R. Tanaś and Z. Ficek, J. Opt. B: Quantum Semiclassical Opt. 6, 90 共2004兲. 关10兴 T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, e-print arXiv:quant-ph/0704.0461. 关11兴 B. Schumacher, Phys. Rev. A 54, 2614 共1996兲. 关12兴 H. Barnum, M. A. Nielsen, and B. Schumacher, Phys. Rev. A 57, 4153 共1998兲. 关13兴 T. Yu and J. H. Eberly, e-print arXiv:0707.3215. 关14兴 H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems 共Oxford University Press, Oxford, 2002兲. 关15兴 U. Weiss, Quantum Dissipative Systems, 2nd ed. 共World Scientific, Singapore, 1999兲. 关16兴 H. P. Breuer, D. Burgarth, and F. Petruccione, Phys. Rev. B 70, 045323 共2004兲. 关17兴 X. Z. Yuan, H. J. Goan, and K. D. Zhu, Phys. Rev. B 75, 045331 共2007兲. 关18兴 L. Tessieri and J. Wilkie, J. Phys. A 36, 12305 共2003兲. 关19兴 C. M. Dawson, A. P. Hines, R. H. Mckenzie, and G. J. Milburn, Phys. Rev. A 71, 052321 共2005兲. 关20兴 J. Jing and H. R. Ma, Chin. Phys. 16, 1489 共2007兲. 关21兴 A. O. Caldeira and A. J. Leggett, Ann. Phys. 共N.Y.兲 149, 374 共1983兲.
关22兴 N. V. Prokofev and P. C. Stamp, Rep. Prog. Phys. 63, 669 共2000兲. 关23兴 G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 共1999兲. 关24兴 C. W. Gardiner, Quantum Noise 共Springer-Verlag, Berlin, 1991兲. 关25兴 S. Shresta, C. Anastopoulos, A. Dragulescu, and B. L. Hu, Phys. Rev. A 71, 022109 共2005兲. 关26兴 S. Paganelli, F. de Pasquale, and S. M. Giampaolo, Phys. Rev. A 66, 052317 共2002兲. 关27兴 M. Lucamarini, S. Paganelli, and S. Mancini, Phys. Rev. A 69, 062308 共2004兲. 关28兴 A. Hutton and S. Bose, Phys. Rev. A 69, 042312 共2004兲. 关29兴 J. Jing and Z. G. Lü, Phys. Rev. B 共to be published兲. 关30兴 Y. Hamdouni, M. Fannes, and F. Petruccione, Phys. Rev. B 73, 245323 共2006兲. 关31兴 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 共1935兲. 关32兴 J. Bell, Physics 共Long Island City, N.Y.兲 1, 195 共1964兲. 关33兴 H. P. Breuer, Phys. Rev. A 69, 022115 共2004兲. 关34兴 N. Canosa and R. Rossignoli, Phys. Rev. A 73, 022347 共2006兲. 关35兴 Z. Ficek and S. Swain, Quantum Interference and Coherence: Theory and Experiments 共Springer, New York, 2005兲. 关36兴 T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 共1949兲. 关37兴 J. Jing and H. R. Ma, Phys. Rev. E 75, 016701 共2007兲. 关38兴 G. Arfken, Mathematical Methods of Physicists, 3rd ed. 共Academic, New York, 1985兲. 关39兴 V. V. Dobrovitski and H. A. De Raedt, Phys. Rev. E 67, 056702 共2003兲. 关40兴 X. G. Hu, Phys. Rev. E 59, 2471 共1999兲. 关41兴 S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 共1997兲. 关42兴 W. K. Wootters, Phys. Rev. Lett. 80, 2245 共1998兲. 关43兴 L. Fedichkin and V. Privman, e-print arXiv:cond-mat/0610756. 关44兴 Z. Gedik, Solid State Commun. 138, 82 共2006兲. 关45兴 T. Yu and J. H. Eberly, Opt. Commun. 264, 393 共2006兲. 关46兴 T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 共2006兲.
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