Entanglement Dynamics of Quantum States Undergoing Decoherence from a Driven Critical Environment

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1 Commun. Theor. Phys. 6 (213) Vol. 6, No. 4, October 15, 213 Entanglement Dynamics of Quantum States Undergoing Decoherence from a Driven Critical Environment MA Xiao-San ( Ò), 1, QIAO Ying (Þ ), 1 LIU Xiao-Dong ( ü), 1 and WANG An-Min ( Ë ) 2 1 School of Electric Engineering and Information, Anhui University of Technology, Ma anshan 2432, China 2 Department of Modern Physics, University of Science and Technology of China, Hefei 2326, China (Received March 18, 213; revised manuscript received June 8, 213) Abstract In this paper, we have investigated the quantum entanglement of quantum states undergoing decoherence from a spin environment which drives a quantum phase transition. From our analysis, we find that the entanglement dynamics depends not only on the coupling strength but also on the external magnetic field and the number of the freedom degrees of the environment. Specially, our results imply that the decay of the entanglement can be enhanced by the quantum phase transition of the environment when the system is coupled to the environment weakly. Additionally, the discussion of the case of the multipartite states with high dimensions is made. PACS numbers: 3.65.Ud, 3.67.Mn Key words: quantum entanglement, entanglement dynamics, quantum phase transition 1 Introduction Entanglement attracts much attention from physicists either in theory or in experiment 1 5] as it plays an important role in quantum information processing such as quantum communications 2] and quantum cryptography. 6] A lot of works concerning entanglement have been done. One of the interesting problems of entanglement is how to measure the extent to which the state entangles, that is entanglement measure. In 1998, Wootters 4] proposed one quantity named concurrence to measure the quantum entanglement of two-qubit states. As is necessary and sufficient for the two-qubit systems, concurrence has been applied to investigate the quantum correlation extensively. The study of the entanglement property is necessary for constructing quantum gates and quantum information processing. Entanglement dynamics is one basic problem and it has been investigated by the researchers with various models of decoherence 7 14] to find some dynamical properties for entanglement evolution of quantum states undergoing decoherence. Different from the above models with a static environment, 7 1] in this paper, we consider the entanglement dynamics of a system undergoing decoherence from a driven critical environment. Quantum decoherence 15 16] is unavoidable because of the interaction between the system and the environment. The extent to which the environment affects the quantum entanglement is an interesting problem and many works have been done along the line. These works mainly focused on the systems coupled to a static environment descried by a time-independent Hamiltonian. In this paper as an extension of the work 17] from a one-qubit system to a two-qubit system, we consider a spin-1/2 system coupled to a driven critical environment which undergoes a quantum phase transition. 18] Different from the results obtained by a static environment, our consideration of a time-dependent environment will present some interesting results as the quantum decoherence is determined by the environment s Hamiltonian and its quantum phase transition. Such a study is necessary as it combines the theory of quantum decoherence with the rapidly growing fields of dynamics of quantum phase transition and quantum information theory. 19 2] Therefore, in this paper, we will make an analysis of the entanglement dynamics of quantum states undergoing decoherence from a driven critical environment. The content is arranged as follows. In Sec. 2, we introduce the model and derive the time evolution of quantum states. In Sec. 3, the main results are presented and analyzed in detail. Finally, we conclude our results in Sec Dynamics of Quantum States In order to get the knowledge of the entanglement dynamics, we should introduce the model and derive the time evolution of quantum states. We consider the system coupled to the environment as a driven Ising spin chain undergoing a quantum phase transition. The Hamiltonian reads N 2 ] H = σi x σx i+1 g(t) N + δ σj z, (1) i=1 j=1 σ z j i=1 where N is the total sites of the environment, g(t) as the Supported by the National Natural Science Foundation of China under Grant Nos , , 1141, and Corresponding author, mxiaosan@mail.ustc.edu.cn c 213 Chinese Physical Society and IOP Publishing Ltd

2 No. 4 Communications in Theoretical Physics 411 driving of the environment is a slow ramp down of a magnetic field and σ x α, σ z α (α = i, j) are the familiar Pauli matrices. The Ising spin chain in a transverse field has been experimentally studied in Refs ]. The initial state of the system is a general pure quantum state ψ s () = c 1 +c 2 1 +c 3 1 +c 4 11 where c 1, c 2,..., c 4 are the coefficients and satisfy the normalization relation 4 i=1 c i 2 = 1. It should be noted that the following calculation holds for the mixed states too. The initial state of the environment is its instantaneous ground state of GS. The composite state ψ is given as ψ () = ψ s () GS. (2) The time evolution of the composite state takes the following expression ψ(t) = ˆT t ] dthg(t)] ψ() = c 1 ˆT exp i t ] dthg(t) + 2δ] GS + c 2 1 ˆT t + c 3 1 ˆT t ] dthg(t)] GS ] dthg(t)] GS + c 4 11 t ] dthg(t) 2δ] GS = c 1 φ 1 (t) + (c c 3 1 ) φ 2 (t) + c 4 11 φ 3 (t), (3) where ( ˆT) is the time-ordering operator, and the evolution of the environmental states coupled to the system s state is given by i t φ x(t) = H x φ x (t). (4) In above equation, the index of x takes the values of 1, 2, and 3. H 1, H 2, H 3 are the corresponding Hamiltonian with the expressions of Hg(t) + 2δ], Hg(t)], Hg(t) 2δ], respectively. The time evolution of the quantum state of the system takes the following expression c 1 2 c 1 c 2 d 1(t) c 1 c 3 d 1(t) c 1 c 4 d 2(t) c ρ s (t) = Tr e ψ(t) ψ(t) = 2 c 1 d 1 (t) c 2 2 c 2 c 3 c 2 c 4 d 3(t) c 3 c 1d 1(t) c 3 c 2 c 3 2 c 3 c 4d 3 (t), (5) c 4 c 1d 2(t) c 4 c 2d 3(t) c 4 c 3d 3(t) c 4 2 where d 1 (t), d 2 (t), d 3 (t) are decoherence factors and take the expressions of d 1 (t) = φ 1 (t) φ 2 (t), d 2 (t) = φ 1 (t) φ 3 (t), d 3 (t) = φ 2 (t) φ 3 (t), and denotes the complex conjugation. The analytical expression of the decoherence factors can be obtained by mapping the spins onto noninteracting fermions with the Jordan Wigner transformation. 17,23] With Bogolubov modes u x k and vx k through φ x (t) = Π k> u x k (t) k, k iv x k (t) 1 k, 1 k ], (6) where k, k, 1 k, 1 k describe the states of the pair quisiparticles with momentum k = (2q + 1)π/N, q =, 1,...,N/2 1. It should be noted that we have set that the value of N is an even number. Therefore, we can obtain the expression D x (t) = d x (t) 2 = Π k> F kx (t) = Π k>,β γ u kβ v kγ + v kβ u kγ 2. (7) It should be noted that when x = 1, the parameters of β, γ take values of 1, 2, when x = 2, the parameters of β, γ take values of 1, 3, and when x = 3, the parameters of β, γ take values of 2, 3, respectively. In order to get the explicit expression of the modes u kβ, v kβ (β = 1, 2, 3), we will consider the adiabatical evolution of the modes. The decoherence factors can be obtained as the squared overlap between the ground states of H x. This implies that v x k(t) = cos(θ x /2), u x k(t) = sin(θ x /2), (8) where θ x, π], cos(θ x ) = e x / 1 + e 2 x, and ε x = Λ x /(sink). Here, the expressions of Λ x, x = 1, 2, 3 read as follows. Λ 1 = g(t) 2δ + cosk, Λ 2 = g(t) + cosk, Λ 3 = g(t) + 2δ + cosk. (9) Therefore, the decoherence factor of Fk x (t) can be obtained as ( Fk x (t) = cos 2 θ β θ ) γ. (1) 2 Therefore, with the reduced density matrix, we can analyze the entanglement dynamics of quantum states undergoing decoherence from a driven environment which exhibits a quantum phase transition. 3 Main Results In order to examine the entanglement dynamics of the system, we introduce the entanglement measure based on concurrence. 4] Here, the entanglement measure based on concurrence is concerned and it is entanglement of formation (EoF). The concurrence is defined by C = max{, λ 1 λ 2 λ 3 λ 4 }, where λ 1, λ 2, λ 3, and λ 4 are the square roots of the eigenvalues in decreasing order, of the matrix R = ρ ρ. Here ρ is the time-reversed matrix ρ = σ 1 y σ 2 yρ σ 1 y σ 2 y and * denotes complex conjugation. Here, we consider that the initial state as a Werner state 24] of the quantum system takes the following expression. ρ s () = 1 p 4 I p Bell Bell, (11)

3 412 Communications in Theoretical Physics Vol. 6 where the parameter of p is used to characterize the noise, I 4 4 is the identity operator with rank of 4, and Bell refers to the Bell state and takes the expression as Bell = (1/ 2)( + 11 ). With the above analysis, we can obtain the time evolution of concurrence as { C(t) = max, pd 2 (t) 1 p }. (12) 2 From the expression, it is easy to find that the larger the parameter of p is, the larger the concurrence is, and the more entangled the state is. This also suggests that the less the noise is, the more entangled the state is. We plot Fig. 1 to demonstrate such a point. From Fig. 1, we can find that the entanglement dynamics depends on the parameter of noise p obviously. The larger the parameter of p is, the larger the concurrence is. For a certain parameter of p, the entanglement takes a slowing-down behavior with the increasing of g(t) from to 1 and then takes an asymptotic rise from to the maximum with the increasing of g(t) from 1 to 2. It should be noted that the point g(t) = 1 is the critical point of the environment. In this sense, we can say that the entanglement is vanishing by the quantum phase transition of the environment. can say only under the weak coupling can the quantum phase transition of the environment enhance the quantum entanglement vanishing of the system. Additionally, the entanglement decreases with the increasing of the coupling constant. The larger the coupling constant is, the smaller the entanglement is. For instance, when g(t) =, the value of entanglement is.95 for the case of δ =.1, while the value of the entanglement is.58,.5,.1 for the cases of δ =.3,.7,.1, respectively. In this point, we can conclude that the strong coupling plays a positive role in shrinking entanglement of the system. Fig. 2 Quantum concurrence versus the driving external magnetic field of g(t) for various coupling constants of δ is plotted, respectively, where p =.99, N = 6. Fig. 1 Quantum concurrence versus the driving external magnetic field of g(t) and the noise parameter of p is plotted, respectively, where δ =.1, N = 6. With the expression of the concurrence, we can analyze the entanglement dynamics in the following content to study the dependence relation of the entanglement dynamics on the coupling constant and the number of the freedom degrees of the environment. From Fig. 2, we find that for the weak couplings such as δ =.1,.3 the quantum entanglement will decrease when the external magnetic field passes through the critical points at which the environment exhibits a quantum phase transition. However, for the cases of δ =.7,.1, the entanglement vanishing could not be enhanced by the quantum phase transition of the environment as it can be seen that the entanglement takes very small values in the range of g(t) with a value between.6 and 1.2. In this sense, we In order to analyze the effect of the number of the freedom degrees of the environment on the entanglement dynamics of the system, we plot Fig. 3. In Fig. 3, we consider two cases of the weak coupling δ =.1 in Fig. 3(a) and the strong coupling δ =.1 in Fig. 3(b). For the weak coupling, the entanglement evolution versus the driven external magnetic field takes similar behaviors for the different numbers of the freedom degrees of the environment. The quantum phase transition of the environment can enhance the entanglement vanishing of the system. However, there is some difference for the behaviors of the entanglement evolution for the different numbers of the freedom degrees of the environment. The smaller the number of the freedom degrees of the environment is, the more obvious the entanglement decay is enhanced by the quantum phase transition of the environment as that the entanglement exhibits a sharp decrease when the external magnetic field takes a value of 1. While for the strong coupling, the entanglement evolution takes different behavior from the weak coupling. As the first observation, the entanglement takes a smaller value for the strong coupling than the weak coupling. This suggests that the strong coupling can shrink the entanglement. For the second observation, there is no obvious evidence to prove that the quantum phase transition of the environment can enhance

4 No. 4 Communications in Theoretical Physics 413 the entanglement decay. Specially, the larger the freedom degrees of the environment is, the larger the range of g(t) in which the entanglement is zero is. In this sense, we can conclude that the quantum phase transition of the environment can enhance the entanglement decay when the system is coupled to the environment weakly. Here, our simulation gives further confirmation of the fact that the quantum phase transition can enhance the entanglement decay of the system when the system is coupled to the environment weakly. Fig. 3 Quantum concurrence versus the driving external magnetic field of g(t) and the number of the freedom degrees of the environment N is plotted for the weak coupling of δ =.1 and δ =.1, p =.99, respectively. Beyond the state considered above, the other states can also be analyzed in detail. For example, when the initial state of the system is ψ = (1/ 2)( ), our results imply that such a state does not perceive the decoherence of the environment. This point is of no surprise because it is an eigenstate of the Hamiltonian. Therefore, it is a decoherence-free entangled state ] In fact, if the parties of the system are coupled to the environment differently, the state of ψ will be affected by the environment. Due to the fact that the entanglement measure of the concurrence is not applicable to the many-qubit or qutritqutrit state, we will not analyze the entanglement dynamics of many-qubit or qutrit-qutrit states in detail here but make some discussion of the results. Our results can also be extended to the multipartite states with high dimensions. The extension to the many-qubit or qutrit-qutrit states of the system is straightforward by the replacement of the eigenvalues of the Hamiltonian with the appropriate values when the states of the system are many-qubit states or qutrit-qutrit states. Specially, the state of a many-qubit system or a qutrit-qutrit system, which is the eigenstate of the system s part of the interaction Hamiltonian, will not perceive the decoherence induced by the environment. However, the state of a many-qubit system or a qutrit-qutrit system, which is not the eigenstate of system s part of the interaction Hamiltonian, will lose entanglement when the environment exhibits a quantum phase transition. With a careful analysis, we find that our results obtained for the two-qubit state can be applied to the many-qubit state or the qutrit-qutrit state to a large extent. The decay of the entanglement of the state for a many-qubit system or for a qutrit-qutrit system with negativity 28 29] as entanglement measure can be enhanced by quantum phase transition of the environment when the system is coupled to the environment weakly. The effect of the noise parameter and the freedom degrees of the environment on the entanglement dynamics of the many-qubit states or qutrit-qutrit states is similar to the case of the two-qubit states analyzed above. 4 Discussion and Conclusions To conclude, we have investigated the entanglement evolution of quantum states of the system coupled to the environment with a driven external magnetic field. With an analysis with the states of concern, we find that the entanglement evolution depends on the noise parameter, the driven external magnetic field, the coupling constant, and the number of the freedom degrees of the environment. The less the noise is, the more the entanglement is. With regards to the coupling constant, we find that the larger the coupling constant is, the smaller the value of the entanglement is. The number of the freedom degrees of environment affects the entanglement evolution too. The larger the number of the freedom degrees of the environment is, the smaller the value of entanglement is. As one special point, the entanglement decay can be enhanced by the quantum phase transition of the environment when the system is coupled to the environment weakly from our simulation. Such a point can provide some clue to the relation of quantum phase transition and the entanglement decay. Our results can also be extended to the states of a many-qubit system or a qutrit-qutrit system. In a word, our study can contribute to some understanding of the entanglement evolution of the quantum states coupled to an environment with a driven external magnetic field.

5 414 Communications in Theoretical Physics Vol. 6 References 1] C.H. Bennett, Phys. Rev. Lett. 68 (1992) ] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 7 (1993) ] A. Peres, Phys. Rev. Lett. 77 (1996) ] W.K. Wootters, Phys. Rev. Lett. 8 (1998) ] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2). 6] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74 (22) ] H.T. Quan, Z. Song, X.F. Liu, P. Zanardi, and C.P. Sun, Phys. Rev. Lett. 96 (26) ] Z. Sun, X.G. Wang, and C.P. Sun, Phys. Rev. A 75 (27) ] Xiao San Ma, An-Min Wang, and Ya Cao, Phys. Rev. B 76 (27) ; S. Maniscalco, S. Olivares, and M.G.A. Paris, Phys. Rev. A 75 (27) 62119; R. Vasile, S. Olivares, M.G.A. Paris, and S. Maniscalco, Phys. Rev. A 8 (29) 62324; Jing Nie, Lin-Cheng Wang, and Xue- Xi Yi, Commun. Theor. Phys. 51 (29) 815; Lan Zhou, Jing Lu, and Tao Shi, Commun. Theor. Phys. 52 (29) ] R. Vasile, S. Olivares, P. Giorda, M.G.A. Paris, and S. Maniscalco, Phys. Rev. A 82 (21) 12312; Yu-Guang Yang, Yuan Wang, Yi-Wei Teng, and Qiao-Yan Wen, Commun. Theor. Phys. 55 (211) 589; Ya Chen and Shi- Qun Zhu, Commun. Theor. Phys. 56 (211) ] M. Genoni, P. Giorda, and M.G.A. Paris, Phys. Rev. A 78 (28) 3233; G. Brida, I. Degiovanni, A. Florio, M. Genovese, P. Giorda, A. Meda, M. G. Paris, and A. Shurupov, Phys. Rev. Lett. 14 (21) ] M. Genoni, P. Giorda, and M.G.A. Paris, Phys. Rev. A 78 (28) 3233; G. Brida, I. Degiovanni, A. Florio, M. Genovese, P. Giorda, A. Meda, M.G. Paris, and A. Shurupov, Phys. Rev. Lett. 14 (21) ] Jian Ma, Zhe Sun, X.G. Wang, and Franco Nori, Phys. Rev. A 85 (212) ] Laleh Memarzadeh and Stefano Mancini, Phys. Rev. A 87 (213) ] D. Giulini, et al., Decoherence and the Appearance of a Classical World in Quantum Theory, Springer (1996). 16] W.H. Zurek, Rev. Mod. Phys. 75 (23) ] Bogdan Damski, H.T. Quan, and Wojciech H. Zurek, Phys. Rev. A 83 (211) ] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, United Kingdom (1999). 19] P. Zanardi and N. Paunkovic, Phys. Rev. E 74 (26) ] M.M. Rams and B. Damski, Phys. Rev. Lett. 16 (211) ] R. Coldea, D.A. Tennant, E.M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327 (21) ] J. Zhang, F.M. Cucchietti, C.M. Chandrashekar, M. Laforest, C.A. Ryan, M. Ditty, A. Hubbard, J.K. Gamble, and R. Laflamme, Phys. Rev. A 79 (29) ] J. Dziarmaga, Phys. Rev. Lett. 95 (25) ] Werner considered only the case p = 1/2. These More General States were Introduced by J. Blank and P. Exner, Acta Univ. Carolinae, Math. Phys. 18 (1977) 3. 25] D.A. Lidar, I.L. Chuang, and K.B. Whaley, Phys. Rev. Lett. 81 (1998) ] D.A. Lidar, D. Bacon, and K.B. Whaley, Phys. Rev. Lett. 82 (1999) ] D. Bacon, D.A. Lidar, and K.B. Whaley, Phys. Rev. A 6 (1999) ] K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58 (1998) ] G. Vidal and R.F. Werner, Phys. Rev. A 65 (22)