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Li Xiao-Ming, Chen Yong-Xu, Xia Yun-Jie, Zhang Qi, Man Zhong-Xiao. Quantum entanglement dynamics based on composite quantum collision model. Chinese Physics B, 2020, 29(6): 060302  
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Quantum entanglement dynamics based on composite quantum collision model
Li Xiao-Ming, Chen Yong-Xu, Xia Yun-Jie †, Zhang Qi, Man Zhong-Xiao ‡
School of Physics and Physical Engineering, Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Qufu Normal University, Qufu 273165, China

 

† Corresponding author. E-mail: yjxia@qfnu.edu.cn zxman@qfnu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 61675115 and 11974209), the Taishan Scholar Project of Shandong Province of China (Grant No. tsqn201812059), and the Shandong Provincial Natural Science Foundation of China (Grant No. ZR2016JL005).

Abstract

By means of composite quantum collision models, we study the entanglement dynamics of a bipartite system, i.e., two qubits S1 and S2 interacting directly with an intermediate auxiliary qubit SA, while SA is in turn coupled to a thermal reservoir. We are concerned with how the intracollisions of the reservoir qubits influence the entanglement dynamics. We show that even if the system is initially in the separated state, their entanglement can be generated due to the interaction between the qubits. In the long-time limit, the steady-state entanglement can be generated depending on the initial state of S1 and S2 and the environment temperature. We also study the dynamics of tripartite entanglement of the three qubits S1, S2, and SA when they are initially prepared in the GHZ state and separated state, respectively. For the GHZ initial state, the tripartite entanglement can be maintained for a long time when the collision strength between the environment qubits is sufficiently large.

PACS: ;03.65.Ud;;03.65.Yz;
Keyword:;entanglement dynamics;;steady-state entanglement;;collision model;
1. Introduction

Quantum entanglement, which is a unique concept of quantum mechanics, is the most important resource of quantum information processing.[1,2] By using quantum entanglement, ones have achieved amazing tasks that cannot be realized by classical means, such as quantum teleportation,[3,4] quantum key distribution,[5] and so on. In practice, however, the noises of various environments usually destroy the useful entanglement even in a finite time, a phenomenon called as entanglement sudden death (ESD).[6–14] Therefore, understanding the dynamics of entanglement[15–19] is not only a fundamental problem but also necessary for the practical applications. The entanglement has also been shown to be closely related to the quantum thermodynamics.[20] It has been proved that the steady-state entanglement can be achieved in the quantum systems coupled to thermal reservoirs.[21,22]

The collision model, an efficient microscopic framework for describing the dynamics of open quantum system, was first put forward by J. Rau in 1963.[23] Recently, it has been used to study and analyze the quantum non-Markovian dynamics[24–31] as well as quantum thermodynamics.[32,33] In particular, the authors in Ref. [28] have investigated an important known instance, which consists of a quantum non-Markovian dynamics system (such as the emission of an atom into a reservoir featuring a Lorentzian spectral density) mapped on to the memoryless composite collision model. In Ref. [33], the authors studied the validity of the Landauer principle in the non-Markovian regime by means of collision models. The collision model is a useful tool for simulating the interaction between open systems and the environment, especially when the intracollisions of the environment particles are considered. Therefore, the collision model can provide a more effective way to study the entanglement dynamics of the open quantum system coupled to thermal environment. In addition, the collision model can vividly demonstrate the mechanism of the non-Markovian process.[30] Because the backflow of information to the system can be simulated by introducing the interaction between the auxiliary particles in the environment. Benefit from its flexibility, the collision model can also be applied to other fields, such as quantum thermodynamics, by changing the initial states of the auxiliary particles and the forms of interactions between particles. We study the entanglement dynamics by using the collision model to simulate the structured environment, such as the electromagnetic field with Lorentz spectral density, which has been studied extensively. The results are consistent with the previous conclusions, which implies the validity of the considered model, and hence one can further apply this model to explore issues which are difficult to be resolved by the usual approaches.

In this work, we investigate the bipartite and tripartite entanglement dynamics by means of the collision model in the presence of intracollisions of environment particles. We first study the entanglement dynamics of the system composed by two qubits interacting with a heat reservoir via an auxiliary qubit. The generation conditions and evolution characteristics of the steady-state entanglement in the long-time limit are studied in detail. In addition, we study the tripartite entanglement dynamics of three qubits which are initially in the Greenberger–Horne–Zeilinger (GHZ) state or the given separated states. Some methods to avoid or suppress ESD during the evolution of the tripartite entanglement dynamics are also presented.

2. Collision model

The system under consideration consists of two qubits S1 and S2 which interact with the thermal reservoir R through the intermediate auxiliary qubit SA. The thermal reservoir R comprises a large number of identical qubits {Rn}. The system qubits, auxiliary qubit, and environment qubits are described by the Hamiltonians , , and , respectively. The ωS, ωA, and ωR are the corresponding transition frequencies of the qubits. For simplicity, we assume ωS = ωA = ωR = ω, and the environment qubits are initially in the same thermal state , with T being the temperature and Z = Trexp (−HR/kB T) being the partition function. In addition, we set ħ = 1 and kB = 1 throughout this paper. The coupling between the system and the environment is assumed to be white noise, i.e., the environment is so large that the auxiliary qubit SA can collide with the same environment qubit only once in the process of interaction with the thermal reservoir. For the appropriate parameters and interactions between environmental qubits, such a model can simulate two two-level atoms in a dissipative cavity (i.e., the Tavis–Cummings model[34]).

The process of interaction between the system and environment is described in Fig. 1. Firstly, pairwise collisions (S1–S2, S2–SA, S1–SA) occur via the unitary time-evolution operator (m,n = 1,2,A, m ≠ n). The collisions induce correlation among them, as shown in Fig. 1(b). Next, the auxiliary qubit SA collides with R1 via the unitary operator , and the correlation among S1, S2, SA, and R1 is established, as shown in Fig. 1(c). Then R1 collides with R2 via the unitary operator , as a result, the correlation among S1, S2, SA, R1, and R2 is established and the first round of collision is completed. After taking the partial trace of the total state, which consists of S1, S2, SA, R1, and R2, with respect to R1, we can obtain the reduced state of S1, S2, SA, and R2, which is the initial state for the next round of collision, and so on in a similar fashion. Therefore, from the reduced state of S1, S2, SA, and Rn + 1 obtained after the n-th round, the total state of S1, S2, SA, Rn + 1, and Rn + 2 after the (n + 1)-th round can be easily obtained, which can be written as follows:

where ρS1S2SARn + 1 is the reduced state of S1, S2, SA, and Rn + 1 after the n-th round collision, ρS1S2SARn + 1 Rn + 2 is the total state of S1, S2, SA, Rn + 1, and Rn + 2 after the (n + 1)-th round collision. So we can obtain the reduced state ρS1 S2 SA Rn + 2 of S1, S2, SA, and Rn + 2 after the (n + 1)-th round collision by tracing out the qubit Rn + 1, namely, ρS1 S2 SA Rn + 2 = TrRn + 1 ρS1S2SARn + 1Rn + 2. In other words, the reduced state ρS1S2SARn + 2 can be expressed through the state ρS1S2SARn + 1. Thus, the iterative relationship between the states ρS1S2SARn + 2 and ρS1S2SARn + 1 after two consecutive collisions is established. The reduced state of the system after the (n + 1)-th collision round can be obtained by tracing out the environment and auxiliary qubit degrees of freedom, namely, .

Fig. 1. The sketch of collision model. (a) Pairwise collisions S1–S2, S2–SA, S1–SA occur between system qubits and auxiliary qubit SA, then they are correlated. (b) Auxiliary qubit SA collides with the first environment qubit R1, the dotted lines in the graph denote the correlation among S1, S2, and SA established after their collisions (i.e., after step (a)). (c) The collision between ancilla qubits R1 and R2, the dotted lines denote the correlation among S1, S2, SA, and R1 established after step (b). (d) The ancilla qubit R1 is traced out, then repeat the process (a)–(c).

As for the interaction describing a collision, many options are available. In our collision model, we select a Heisenberg-like coherent interaction, namely,

where is the Pauli operator, and gAB is the coupling constant between qubits A and B. The collisions between the two qubits can be described by a unitary operator , where τ is the collision time. Here, by means of the conclusion[35]

where is the identity operator, and is the two-qubit swap operator with the action |φ1⟩ ⊗ |φ2⟩ → |φ2⟩ ⊗ |φ1⟩ for all |φ1⟩, |φ2⟩ ∈ ℂ2. We can express the unitary time evolution operator as

where χAB = 2gABτ (0 ⩽ χAB ⩽ π/2) is the dimensionless interaction strength between qubits A and B, with (A,B) = (S1,S2), (S1,SA), (S2,SA), (SA,Rn), (Rn,Rn + 1). In the ordered basis vectors {|00⟩AB,|01⟩AB,|10⟩AB,|11⟩AB}, the unitary operator can be written as

For convenience, we let (χS1S2,χS1SA,χS2SA,χSARn,χRnRn + 1) = (κ,λ1,λ2,M,Γ) in the following.

3. Bipartite quantum entanglement

For two-qubit quantum systems, there are many methods to measure the quantum entanglement, such as the concurrence proposed by Wootters[36,37] and the negativity.[38] In this paper, we use concurrence to measure the bipartite entanglement. If the density matrix of a two-qubit quantum system can be expressed in the following form:

the concurrence of the system can be expressed as

3.1. Dynamics of bipartite entanglement

In this section, we assume that the system is initially in an entangled state with the form

where and the auxiliary qubit is initially in the same thermal state as the environment qubits.

Firstly, we consider the case that there is no interaction between S1 and S2, and between S1 and SA, i.e., κ = 0 and λ1 = 0. If the appropriate parameters and interactions are selected, this model resembles the dissipative Jaynes–Cummings model (SA corresponds to a lossy cavity mode, interacting with the system S2. S1 is outside the cavity and does not interact with others). The concurrence CS1S2 as a function of collision rounds n for different Γ is shown in Fig. 2. It should be noted that the time after n rounds collisions is t = nkτ (k is the number of collisions in each collision round), that is to say, the collision rounds n is proportional to the time. Therefore, the collision rounds n represents the evolution time of the entanglement of the system. One can see that the entanglement can partially revive after sudden death. The concurrence CS1S2 of the two qubits S1 and S2 gradually decreases with Γ, and the larger the collision strength Γ is, the faster the entanglement decreases until the entanglement disappears completely. In fact, different from the usual dissipative Jaynes–Cummings model, here SA is coupled to the non-Markovian environment due to the intracollision of the environmental qubits. Therefore, the revival of the concurrence CS1S2 is inversely proportional to the intracollision strength Γ rather than being proportional to it. Furthermore, a comparison between Figs. 2(a) and 2(b) shows that the larger the values of λ2 and M, the faster the oscillation of CS1S2 during the evolution. In the long-time limit, the entanglement will disappear completely, namely, the steady entangled state cannot be achieved, regardless of the choice of the initial states, as shown in Fig. 2(c).

Fig. 2. (a), (b) Concurrence of the system as a function of collision rounds n for different intracollision strengths Γ with θ = π/4. (c) Concurrence as a function of collision rounds n for different initial states with Γ = 1.2, λ2 = 0.05, and M = 0.05. The other parameters are T = 1, κ = λ1 = 0, and ω = 1.

Next, we discuss the entanglement evolution characteristics of the system taking into account the interaction between S1 and S2 and the interaction between S1 and SA (i.e., κ,λ1 ≠ 0). In Fig. 3, we show the entanglement against the collision rounds n for different intracollision strength Γ. We can see that even S1 and S2 are initially in a separate state (i.e., θ = 0, as shown in Fig. 3(a)), the interactions of the two qubits can entangle them, but it cannot reach the maximum entanglement. In this case, the concurrence CS1S2 of the system shows an oscillation behavior, and there is no ESD during the evolution of entanglement. In addition, the numerical results show that when collision rounds n is very large, the concurrence CS1S2 tends to be constant, that is to say, the steady-state entanglement arises in the long-time limit with the chosen parameters. When θ = π/12, one can observe that the ESD arises during the evolution of entanglement. Although there exists an increase of the concurrence CS1S2 during the evolution of the entanglement, even exceeding its initial value, as shown in Fig. 3(b), the calculation results show that the entanglement will eventually disappear completely in the long-time limit, and the larger the intracollision strength Γ is, the faster entanglement disappears. That is to say, in this case, the steady-state entanglement does not arise.

Fig. 3. Concurrence of the system as a function of collision rounds n for different intracollision strengths Γ. The other parameters are T = 1, κ = 0.01, λ1 = λ2 = 0.01, M = 0.01, ω = 1 in both plots.
3.2. The bipartite steady-state entanglement

In order to further study the dynamic behavior of entanglement in the long-time limit, the concurrence of S1 and S2 is calculated for different parameters. The numerical results show that the steady-state entanglement will occur in the long-time limit if the parameters are chosen properly. In Fig. 4(a), we show the concurrence for different initial states when T = 0.5, M = 0.05, and κ = λ1 = λ2 = 0.01. It can be seen that the formation of steady-state entanglement depends on the selection of the initial state. There is a threshold θ0, and only when θ < θ0, the steady-state entanglement will arise in the long-time limit. It is noteworthy that θ0 depends on the environmental temperature T. To illustrate this point more clearly, we show a comprehensive picture for the dependence of the steady-state concurrence CS1S2 on T and θ in Fig. 5, where we can see the thresholds θ0 for a given T. Moreover, we also can find from Fig. 5 that the lower the environmental temperature is, the greater the value of θ0 is. It is important to note that for θ = −0.25π, the entanglement of the two qubits S1 and S2 hardly changes and remains at the maximum, as shown by the red dash line in Figs. 4(a) and 5. The steady-state entanglement is greatly affected by the environment temperature. We can see from Fig. 4(b) that the steady-state concurrence CS1S2 decreases with the increase of the environment temperature, if the temperature is too high, the steady-state entanglement will not arise. For a given initial state, the maximum steady-state entanglement can be realized when T = 0. However, it is interesting that in the temperature range where the steady-state entanglement can be formed, the high temperature speeds up the formation of the steady-state entanglement, as shown in Fig. 4(b). The effect of the interaction strengths Γ, M, κ, λ1, λ2 on the steady-state entanglement is shown in Figs. 4(c) and 4(d). It should be noted that only the weak system–environment coupling is considered here. For the relatively small values of Γ, we can see from Fig. 4(c) that increasing the intracollision strength Γ can speed up the formation of the steady-state entanglement. If Γ is further increased after it reaches a certain value Γ0 (with the chosen parameters, Γ0 = 1.3), the formation of the steady-state entanglement will be delayed. In addition, increasing the collision strength M can also speed up the formation of the steady-state entanglement, as shown in Fig. 4(d). However, the steady-state concurrence CS1S2 of the system trends to be the same with the increase of collision rounds n for different collision strengths Γ and M, in other words, for a given initial state and environment temperature, the collision strengths Γ and M (except M = 0) have little effect on the steady-state concurrence CS1S2 of the two qubits S1 and S2. This can be understood by the fact that in the long-time limit, the exchange of information between the system and the environment reaches a balance. In particular, when M = 0, namely, there is no interaction between the auxiliary qubit and the environment, we show the concurrence CS1S2 of S1 and S2, the concurrence CS1SA of S1 and SA, and the concurrence CS2SA of S2 and SA in Fig. 6. It can be seen that these three can be transferred to each other, the evolution of entanglement of the system (S1 and S2) is a periodic process of entanglement transfer and recovery. Therefore, in this case, the steady-state entanglement cannot be generated. Moreover, if the interaction strengths κ, λ1, λ2 among S1, S2, and SA are relatively large (e.g., κ = λ1 = λ2 = 0.05, as shown by the green short dash line and the violet dotted line in Fig. 4(d)), the steady-state entanglement cannot arise. Therefore, we can obtain a steady entanglement by choosing appropriately the initial state, the environment temperature, and the interaction strengths among S1, S2, and SA.

Fig. 4. (a) Concurrence of the system for different initial states with T = 0.5, Γ = 1.2. (b) Concurrence of the system for different temperature T with θ = 0, Γ = 1.2. (c) Concurrence of the system for different intracollision strengths Γ with θ = π/40, T = 0.5. The other parameters are κ = λ1 = λ2 = 0.01, M = 0.05, ω = 1 in the three plots. (d) Concurrence of the system for different collision strengths κ, λ1, λ2, M with θ = π/40, T = 0.5, Γ = 1.2, ω = 1.
Fig. 5. Contour plot of the steady-state concurrence CS1S2 for different T and θ. The other parameters are κ = λ1 = λ2 = 0.01, M = 0.05, Γ = 1.3, ω = 1.
Fig. 6. Concurrence against the collisions rounds n for the initial state θ = −π/4, the other parameters are T = 0.5, κ = 0.01, Γ = 1.2, λ1 = λ2 = 0.01, M = 0, ω = 1.
4. Tripartite quantum entanglement

Now, we study the dynamics of tripartite quantum entanglement.[39–41] We consider the system S composed of S1, S2, and SA. SA interacts with the thermal reservoir through collisions, and initially, there is no correlation between S1 (S2) and the environment. The tripartite entanglement among S1, S2, and SA is analogy to the entanglement among two two-level atoms and a cavity mode.

We adopt different methods to measure the tripartite quantum entanglement for different initial states. When the reduced density matrix describing the system is X-form (i.e., only the diagonal and antidiagonal elements of the matrix are nonzero), we adopt the genuinely multipartite concurrence[42] CGM to measure tripartite entanglement.

In our model, when the system (i.e., three qubits S1, S2, and SA) are initially prepared in the GHZ state, the reduced density matrix of the system is always X-form during the evolution process, which can be written as

where , i = 1, 2, 3, 4. The genuinely multipartite concurrence of ρS1S2 SA is defined as

where (i, j = 1,2,3,4). For the initial state |Ψ⟩ |Φ(0)⟩ ⊗ |φ⟩SA, we adopt negativity N(ρ) to measure the entanglement. Here |φ⟩SA is the initial state of qubit SA, which has the following form:

The negativity N(ρ) is defined as

where ρT is the partial transpose matrix of ρ, and .

4.1. Dynamics of tripartite entanglement for GHZ initial state

We show the genuinely multipartite concurrence CT − GM of ρS1S2SA with GHZ initial state for different intracollision strengths Γ in Fig. 7. When Γ is relatively small, the genuinely multipartite concurrence CT−GM decreases monotonically with the collision round n, and the larger the intracollision strength Γ is, the faster the entanglement disappears. For the sufficiently large intracollision strength Γ (e.g., Γ = 1.55 and Γ = 1.56 in Fig. 7(b)), the evolution of entanglement shows an oscillation behavior until it disappears completely, and no ESD arises in the whole process. This is due to that with the increase of the intracollision strength, the non-Markovianity of the system will be enhanced, which leads to the long-time oscillation of the entanglement before it disappears. In this case, the entanglement of the system can be maintained in a long time. When Γ = π/2, the states of the two reservoir ancillas Rn and Rn + 1 are completely swapped after the collision,[43] the ancilla Rn will return to its original state (i.e., thermal state), while the information of Rn is completely delivered to the ancilla Rn + 1. That is, the system interacts with a fresh reservoir ancilla every time which completely carries the information from the recent previous collision. Therefore, the evolution of entanglement of the system presents perfect periodicity, as shown by the black solid line in Fig. 7(b).

Fig. 7. The genuinely multipartite concurrence of the system for GHZ initial states and different intracollision strengths with M = 0.01, κ = λ1 = λ2 = 0.15, T = 1.
4.2. Dynamics of tripartite entanglement for initial state |Ψ⟩ = |Φ(0)⟩ ⊗ |φ⟩SA

In Fig. 8, we show negativity N(ρS1S2 − SA) as a function of collision rounds n for different initial states. We find that, due to the interactions among S1, S2, and SA, the tripartite entanglement can be generated even though they are initially in the separated state |010⟩S1S2SA (i.e., θ = 0, α = 0). But the numerical results show that the interactions cannot entangle three qubits for some particular initial states of the system, such as |000⟩S1S2SA. When M = 0, as shown in Fig. 8(a), the evolution of negativity N(ρS1S2 − SA) shows a periodic oscillation between the maximum and zero. In this case, the numerical results show that the system evolves into a tripartite entanglement W state for the initial state |Ψ⟩ = |Φ(0)⟩ ⊗ |0⟩SA (i.e., α = 0). If there is a weak interaction between the system and the environment, as shown in Fig. 8(b), N(ρS1 S2 − SA) always exhibits an oscillatory behavior with amplitude attenuation for different initial states with the chosen parameters. In addition, we study the effects of the interaction strengths between the system and the thermal reservoir and the intracollision strengths on the tripartite entanglement dynamics, as shown in Figs. 9 and 10. One can immediately observe from Fig. 9 that the relatively strong interaction between the system and the environment not only speeds up the decay of entanglement but also causes the ESD during the evolution of entanglement. When Γ is relatively small, the evolution of the entanglement shows the oscillation behavior of amplitude attenuation, until it disappears completely, and the smaller the Γ is, the slower the entanglement decays, as shown in Fig. 10(a). For a very large Γ (e.g., Γ = 1.56), the entanglement experiences successive decrease and recovery behaviors, as shown in Fig. 10(b). These results strongly indicate that we can choose the appropriate parameters to obtain the tripartite entanglement which can be maintained for a long time and avoid or reduce ESD in the evolution of entanglement.

Fig. 8. Negativity of the system as a function of collision rounds n for different initial states. The other parameter parameters are κ = λ1 = λ2 = 0.05, Γ = 1.2, T = 1.
Fig. 9. Negativity of the system against collision rounds n for different coupling strength M. The parameters are κ = λ = γ = 0.05, ω = 1, T = 1, Γ = 1.2, α = 0, θ = π/4.
Fig. 10. Negativity of the system against collision rounds n for different intracollision strength Γ. The parameters are θ = π/4, α = 0, κ = λ = γ = 0.05, ω = 1, T = 1, M = 0.01.
5. Conclusions

We study the entanglement dynamics by mean of composite quantum collision models. For the bipartite quantum entanglement, when the two qubits are decoupled to each other and one of the qubits does not interact with the auxiliary qubit, the ESD arises in the process of entanglement evolution. In this case, the steady-state entanglement can not be obtained. If the interaction between the two system qubits and the interaction between each system qubit and the auxiliary qubit are considered, when the system is initially in the separated state, the ESD does not arise, and the steady-state entanglement arises in the long-time limit with the appropriate parameters. In this case, we show the emerging conditions of the steady-state entanglement during the evolution of entanglement and the factors that affect the steady-state concurrence.

We also investigate the tripartite entanglement dynamics among the three qubits S1, S2, and SA for different initial states. For the GHZ initial state, there exists a critical value of the intracollision strength, which makes the tripartite entanglement decay fastest. If the initial state is |000⟩S1S2SA, the three qubits entanglement is not realized by collision. When the system is decoupled to the environment, a tripartite entanglement W state can be achieved for the initial state |Ψ⟩ = |Φ(0)⟩ ⊗ |0⟩SA. In addition, we also show that the weak interaction between the system and the environment can provide a candidate scheme to prepare the tripartite entanglement states that immune to the ESD for initial state |Ψ⟩ = |Φ(0)⟩ ⊗ |φ⟩SA.

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