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Quantum entanglement transfer through two XXZ spin chains
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Quantum entanglement transfer through two XXZ spin chains Ying-Hua Ji a,c,∗ , Yong-Mei Liu b a
Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China c Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China b
a r t i c l e
i n f o
Article history: Received 25 April 2014 Accepted 2 June 2015 Available online xxx Keywords: Quantum communication Quantum entanglement Dzyaloshinsky – Moriya interaction Three-site interaction Phase shift
a b s t r a c t We investigated the quantum entanglement through two independent spin chains with phase transfer √ shift. The initial target states 01 + 10 / 2 or its mixed state is shown to be more suitable for quantum correlation transfer in this system. Generating a phase shift by the Aharonov – Casher effect partly improves the amplitude of the entanglement, whereas it is unfavorable for prolonging the survival time of quantum entanglement. We compared and analyzed the effects of spin chains parameters on quantum discord and quantum entanglement. The result indicated that the quantum entanglement is more robust than quantum discord in the two parallel XXZ spin chains. © 2015 Elsevier GmbH. All rights reserved.
PACS: 03.67.Hk 03.65.Ta 75.10.Pq
1. Introduction Transferring quantum information reliably and efficiently is an important task in quantum information processing [1–3]. Most common approaches to this task include methods with an information bus, guided ions, flying photons, a sequence of swapping operations between the neighboring qubits, etc. However, these methods require additional complexity in structures, manipulations, and controls of the interaction between qubits, as well as the repeated conversions between the qubit state and another physical degree of freedom. Spin chains have been proved to be efficient quantum channels for short distance quantum communications and much effort has been devoted to explore efficient schemes to transfer quantum information perfectly or to enhance the quality effectively [4,5]. It has been reported that perfect state transfer through spin networks of arbitrary length can be achieved by preengineered qubit couplings, local measurements on the individual spins [6], global pulses [7], etc. Abundant and reliable information resource can provide the perfect transfer of information with important protection.
∗ Corresponding author at: Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China. Tel.: +86 13755658129. E-mail address: [email protected] (Y.-H. Ji).
Quantum entanglement has been considered as a valuable resource and it plays a central role in various quantum information processes [8]. However, decoherence is the most important limit for quantum computation because its effect is that the quantum superposition decays into statistical mixtures [9,10]. It is known that the coherence and disentanglement are closely connected. Therefore, an increasing interest to study the quantum mechanism is the entanglement dynamics in quantum computation and quantum information. In 2004, it was found that, when the two-level atoms with an initially entangled state interact respectively with a separate vacuum bath, the entanglement between atoms completely disappears after a short time which is much shorter than that of decoherence. This phenomenon was called as the entanglement sudden death (ESD) [11]. The previous results show that ESD is sensitive to the initial conditions of the qubits and the interacting environments. In order to ensure the correct computation of quantum logic and quantum computation, researchers have done many theoretical and experimental studies on the phenomenon of ESD and shown how to avoid ESD and produce the scheme of sudden birth for some cases during the past decade [12]. How to generate and transfer entanglement is a problem of fundamental interest in quantum information processes [13]. In order to overcome the influence of decoherence, one of the tactics proposed by the electrical engineers and researchers is to improve the decoherent time of the qubits by optimizing the system [14].
http://dx.doi.org/10.1016/j.ijleo.2015.06.008 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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Intensive study on the internal and external factors that influence the dynamical behavior of the system is the prerequisite for optimizing the design of quantum system. Recently, there is a growing interest in the study of spin chains with extra couplings, such as Dzyaloshinsky – Moriya interaction, three-site interaction, phase shift, and energy current [15,16]. The results indicate that the teleportation fidelity of thermal entanglement can be enhanced by Dzyaloshinsky – Moriya interaction, and the maximum discord can be enhanced by the phase shift produced by Aharonov – Casher effect [17]. In spite of many works in the area of quantum information transferring, to the best of our knowledge, there are two interesting questions which remain open: (i) What happens to the quantum entanglement transfer in the spin chains? (ii) Which kind of initial quantum state should be prepared so that the ESD can be effectively avoided and then the quantum information can be transferred continuously through two parallel spin chains of arbitrary length? So, in this paper, we present a detailed analysis on quantum entanglement transfer through two parallel XXZ spin chains.
(2) where, ϕ is the phase change between the neighboring spins, which is produced by the Aharonov – Casher effect. ˆ n˛ (˛ = x, y, z; n = 1, 2, . . ., N) is the Pauli matrix for the site n and respectively satisfies the commutation and anti-commutation algebraic relation, ± = ( x ± i y )/2. The first and second terms are the coupling ones. J is the coupling coefficient in x and y directions and Jz is the coupling coefficient in z direction. This model is reduced to the isotropic XX model when Jz =0 and to the isotropic XXX model when Jz = J. The ˛ periodic boundary conditions N+1 = 1˛ are assumed. For the case of the open chain, the phase shift has no effect on the energy spectrum and therefore causes no change in the quantum entanglement transfer. In order to diagonalize the projected Hamiltonian, we first apply the Jordan – Wigner transformations to map spin to onedimensional spinless fermions with creation and annihilation operators as follows [19]: (3)
l=1
n−
=
k−1
1 − 2a+ a l l
H (j) =
(8)
k>0
with
(j)
Ek = J cos (k + ϕ) +
(N − 4)JZ (N − 2)h , − 4 2
other
= 02 . . .0N
(AB)
(j)
(AB)
(AB) (0) = 1−1 (0) ⊗ other (0),
(AB)
(10)
0. The initial general state 1−1 (0) where, other (0) = 0 0ther other of the first spin pair can be written in Bloch representation (AB)
(AB)
(AB)
(AB) 1−1 (0)
1 = 4
(AB) I1−1
+
3(A) b4 1
⊗ I + b5 I ⊗
3(B) 1
+
3
˛(A) b˛ 1
⊗
˛(B) 1
,
˛=1
(11)
(AB)
where, b˛ = Tr 1−1 ( ⊗ )
(AB)
for ˛ = 1, 2, 3 and I1−1 is the
operator in the subspace of the first identity spin pair. b4 = 3(A)
1
⊗I
(AB)
, b5 = Tr 1−1
3(B)
I ⊗ 1
are the compo-
(AB) (t) = U(t)(AB) (0)U + (t),
(12)
where, U(t) = exp(− iHt). Following the procedures presented in Ref. [20], we can obtain the reduced density matrix for spins n and n’ by tracing over all other degrees of freedom nn (t) as
⎛
11
⎜ ⎜ 0
0
0
14
22
23
0
⎝ 0
32
33
0
0
n−n = ⎜ ⎜ (AB)
41
⎞
⎟ ⎟ ⎟. ⎟ 0 ⎠
(13)
44
The diagonal elements of the reduced density matrix (t) for the chain system are: 11 (t) = 1 −
2 1 + b3 + b4 + b5 2 + b4 + b5 fn (t)4 , fn (t) + 2 2
(14)
22 (t) =
2 1 + b3 + b4 + b5 1 + b5 fn (t)4 , fn (t) − 2 2
(15)
33 (t) =
2 1 + b3 + b4 + b5 1 + b4 fn (t)4 , fn (t) − 2 2
(16)
44 (t) = 1 − [11 (t) + 22 (t) + 33 (t)] , an ,
(9)
Assume that initially the first spin pair is prepared in the general (AB) state 1−1 (0) and the others are set in the ferromagnetic ground
(4)
and the nondiagonal elements are
(5)
14 (t) =
l=1
nz = 2a+ n an − 1.
Ek ck+ ck ,
nents of the local Bloch vectors. The time evolution of the total density of the system is given by
n=1
1 − 2a+ a , l l
where, k = 2m/N with −N/2 ≤ m ≤ N/2. Under the one-magnon condition, the Hamiltonian (1) has the following diagonal form
(AB)
1 iϕ x x y y z = J e n n+1 + e−iϕ n n+1 + Jz nz n+1 + 2B nz , 4
n+ = a+ n
(7)
n
Tr 1−1
N
1 ink + ck+ = √ e an , N
(1)
where, H(j) (j = A, B) denotes the Hamiltonian of the jth XXZ spin chain. Considering the effect of the phase shift, the transformed H(j) can be expressed as
n−1
(6)
n
(j)
It is well known that both the Aharonov – Casher effect and the Dzyaloshinsky – Moriya interaction can generate the phase shift in a spin chain, and this phase will manipulate the quantum information transfer. In this paper, the model we consider consists of two parallel XXZ spin chains with the effect of phase shift. Each Heisenberg chain contains N qubits interacting ferromagnetically with their nearest neighbors. The total Hamiltonian of the system can be written as (kB = = 1)[18]
H (j)
1 −ink ck = √ e an , N
state 0
2. Parallel XXZ spin chains with phase shift
H = H A ⊗ I (B) + I (A) ⊗ H (B) ,
Subsequently, the Fourier transforms of the fermionic operators are described by
b3 − b4 2 fn (t)e2iE0 t , 2
(17)
(18)
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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2 b3 + b4 fn (t) , 2
23 (t) = (j) fn (t)
(19)
(20)
k
(A)
∗ (t). In above equations, we have set f (t) = f and jk (t) = kj n n (t) = (B)
fn (t) and E0 = − N/4. Then we can investigate the quantum correlation between spins n and n’ analytically. 3. Measurement of quantum entanglement Characterization of entanglement is one of the key features related to quantum information theory. From the point of view of a possible application, it is important not only to determine whether a given state is entangled, but also to quantify the degree of entanglement. Among several such quantities, the concurrence introduced by Bennet et al. is often used for this purpose. The concurrence C in a bipartite state is defined as [21]
1 −
2 −
3 −
4 },
(21)
where, j are the eigenvalues of the non-Hermitian W = (t)(t) ¯ arranged in descending order 1 ≥ 2 ≥ 3 ≥ 4 . Spin-flip operator (t) ¯ is defined as (t) ¯ =
(1)
(2)
y ⊗ y
∗ (t) y ⊗ y (1)
(2)
.
The asterisk *(t) is the complex conjugate of density matrix (AB) (t). For this n−n (t), the concurrence will be
C(t) = 2 max{0, 23 (t) − −
11 (t)44 (t), 14 (t)
22 (t)33 (t)},
(22)
which is obviously dependent on the initial conditions. It is indisputable that certain amplitude and duration are necessary for quantum communication and quantum computation. Therefore, this section will focus on the evolution characteristics of the entanglement dynamics of the spin in each position and accordingly determine if the quantum information can be transferred efficiently. We here consider the X-states, which may arise in a wide variety of physical situations, as the target state to be transferred to investigate the transfer properties of quantum entanglement. X-states are also encountered as eigenstates in all the systems with odd-even symmetry like in the Ising and the spin chain models. X-states are defined as:
r, = r (ϕ)
˚ r, = r ˚()
(ϕ) +
˚() +
2
Cn−n = fn (t)
r 1 − cos 2 − 1 + r cos 2
1 (j) = exp ik (n − 1) − iEk t , N
C(t) = max{0,
3
1−r I4 , 4
(23a)
1−r I4 . 4
(23b)
− 1 − r + 4r cos2
2
˚ = fn (t) Cn−n
fn (t)2 ,
r 1 − cos 2
(24)
2 4 − (1 − r) 2 − 2 fn (t) + (1 − r) fn (t) .
(25)
These equations are quite general since their forms are independent on the particular sites of the spin chains, but only on the Hamiltonian model of Eq. (1) and on the chosen initial state. Once the parallel XXZ spin chains structure are specified, the explicit form of fn (t) is obtained and in turns the explicit time-dependence of concurrence is determined. 4. Quantum entanglement transfer In this section we investigate the transfer properties of two entangled qubits transferring through two identical uncoupled chains A and B of length N, each of which by the Hamil is modeled tonian (2). For the initial target state r, , we plot the quantum entanglement that transfers from the first sites of the dual chains to the (n,n) th sites as a function of rescaled time Jt in Fig. 1 and Fig. 2. Figs. 1 and 2 clearly show that the death and rebirth happens to concurrence with the variation of Jt in the target state r, . We also find that for smaller N, quantum entanglement can be transferred from the first spin pair with and without the phase shift to the nth any other spin chain for various (N,n). There are some common features for the concurrence. First of all, for a fixed n, we find that value of the concurrence of nth spin chain decreases with increasing N. Second, in general, entanglement death occurs more easily and entanglement rebirth more difficultly as the transmission distance increases (namely, n increases). Finally, based upon huge numerical computations, we find that when N is fixed, the transmission properties are not obviously improved by manipulating the coupling coefficients and external magnetic intensity. Comparing Figs. 1 and 2, however, the quantum entanglement can be enhanced by a nonzero phase shift for most time, which agrees well with the results in the reference. Particularly, Figs. 1–4 reveal that the phase shift just partly improves the amplitude of
In Eq. (23), r is the purity of the initial states, which ranges from 0 for maximally mixed states to 1 for pure states, I4 is the 4 × 4 identity matrix and
() = cos 00 + sin 11 , ˚() = cos 01 + sin 10 ,
are the Bell-like pure state, and the parameter is sometimes called as the degree of entanglement. The explicit expressions at time t are given for of concurrences two initial states r, and ˚ r, of Eq. (22), respectively, by
Fig. 1. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaledtimeJt without phase shift. The initial state of
the first spin pair is prepared in
r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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Fig. 5. The quantum entanglement that transfers from sites (1, 1) to sites (N − 1, N − 1) as a function of rescaled time Jt with phase shift ϕ = /3. When the chain length: (a) N = 15, (b) 31. The initial state of the first spin pair is prepared in Bell state with JZ = 0.1J, B = J.
Fig. 2. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaled time Jt with phase shift ϕ = /3. The initial state of the first spin pair is prepared in
r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
entanglement and does the maintenance of entanglement no good. In addition, the transfer quality can be enhanced by increasing the purity of the initial states. We also examine the case of the initial target state ˚ r, as shown in Fig. 3 and Fig. 4. By comparing Figs. 3 and 4 with Figs. 1 and 2, it is interesting to find that the effects of phase are very similar for state shift on the quantum entanglement r, and state ˚ r, . The transfer quality for the r,
case is always lower than that for the ˚ r,
case. If the ini-
tial target state is ˚ r, , in the evolution of entanglement, once ESD occurs to the concurrence, almost simultaneously the entanglement rebirth appears, which is beneficial to the information transmission. When the spin chains are larger, as shown in Fig. 5, N = 31, excellent entanglement quality can still be obtained inposition (30,30). Inversely, if the initial target state is r, , the entanglement in the same position is quite small and rebirth is difficult for the entanglement, the evolution quality is poor. Further numerical computations indicate that using the initial target state ˚ r, is more suitable for quantum correlation transfer in this system. Moreover, the transmission quality can be improved obviously via enhancing the purity of the initial state. 5. Quantum discord transfer
Fig. 3. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaledtimeJt without phase shift. The initial state of
the first spin pair is prepared in ˚ r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
Fig. 4. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaled time Jt with phase shift ϕ = /3. The initial state of the first spin pair is prepared in ˚ r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
It has been realized that entanglement represents only a special kind of correlations and there exist other nonclassical correlations such as quantum discord first introduced in Ref. [22]. In many models, it has been reported both theoretically [23] and experimentally [24] that the quantum discord is non-zero even for unentangled states and more robust than the entanglement against sudden death [25], which indicates that quantum information processing based on quantum discord may be more robust than that based on entanglement. However, the quantum discord of a general bipartite state is not always larger than the entanglement, which means that the quantum discord is not simply the sum of entanglement and some other non-classical correlation. The relation between entanglement and quantum discord is complicated for general states. For the pure entangled states, quantum discord coincides with the entropy of entanglement. However, these two measurements disagree on the quantum correlations of a mixed state because the quantum discord may be nonzero for some separable states. The entanglement is more robust than quantum discord in some systems. Therefore, choosing entanglement or discord in a determined quantum system as the information resource remains an important and interesting question. In this section, we simply investigate quantum discord transfer through two identical uncoupled chains A and B of length N, each of which is modeled by the Hamiltonian (1). Currently, how to measure quantum discord is still a hot topic. In order to study the relation between entanglement and quantum discord, we select geometric measure of quantum discord (GMQD)
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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are more suitable for quantum correlation transfer in this system. We compare and analyze the effects of spin chain parameters on the quantum discord and quantum entanglement. The result indicates that the quantum entanglement is more robust than quantum discord in the two parallel XXZ spin chains. Acknowledgment
Fig. 6. The quantum correlation that transfers from sites (1,1) to sites (N − 1, N − 1) chain length (a) as a function of rescaled time Jt with phase shift ϕ = /3. Whenthe /√2. N = 11, (b) 21. The initial state of the first spin pair is prepared in 01 + 10
as the physical quantity for measuring quantum correlation. The GMQD, a measurement of quantum correlations in a quantum system, is defined as [26]
2
DA () = min − , g
∈˝0
(26)
where, the minimum is larger than all possible classical states of the form p1 ˚1 ˚1 ⊗ 1 + p2 ˚2 ˚2 ⊗ 2 with p1 + p2 = 1.
˚1 and ˚2 are two orthogonal bases of the subsystem A. 1 and 2 2 are two density matrices of the subsystem B. Here, − =
Tr( − )2 represents the square norm of the Hilbert – Schmidt space. In Fig. 6, we plot the dynamics of quantum and√entan discord glement for the for the initial target states 01 + 10 / 2 as a function of the scaled time Jt. Fig. 6 shows that the common feature of the dynamics for both of them is a decrease with increasing N. However, the entanglement has stronger robustness than quantum discord. The characteristics of quantum discord transfer are qualitatively different from that of entanglement. We clearly find that: (1) As a whole, the quantum discord is very weak and easily tends to zero in a short time, which is of no practical use. (2) Entanglement survives longer for a fixed N, and the quantum discord death is faster on the contrary. (3) The particularly important difference is √ that: for the initial target states 01 + 10 / 2, it is difficult for the quantum discord to rebirth after the death appears, and that more greater N, more difficult to rebirth. By contrast, rebirth can occur fast to the quantum entanglement as addressed above, which verifies again that the quantum entanglement is qualitatively different from quantum discord. 6. Conclusions We have investigated how the length N or site n of two parallel XXZ spin chains and the Aharonov – Casher effect influence the quantum entanglement when the system is initially in X-state. We find that the time evolution of quantum entanglement is oscillating, and collapse and revival randomly occur. With the increasing of N, the quantum entanglement was quite small and the duration was too short to be used as the quantum information resource. Numerical simulations indicate that the phase shift merely partly improves the amplitude of entanglement and is disadvantageous to We also find prolong the survival time ofquantum entanglement. √ that the initial target states 01 + 10 / 2 or in its mixed state
This work is supported by the National Natural Science Foundation of China (Grant no. 11164009). References [1] S. Bose, Quantum communication through an unmodulated spin chain, Phys. Rev. Lett. 91 (2003) 07901. [2] G.F. Zhang, Thermal entanglement and teleportation in a two-qubit Heisenberg chain with Dzyaloshinski – Moriya anisotropic antisymmetric interaction, Phys. Rev. A 75 (2007) 034304. [3] J. Allcock, N. Linden, Quantum communication beyond the localization length in disordered spin chains, Phys. Rev. Lett. 102 (2009) 110501. [4] V. Subrahmanyam, Entanglement dynamics and quantum-state transport in spin chains, Phys. Rev. A 69 (2004) 034304. [5] C.A. Bishop, Y.C. Ou, Z.M. Wang, M.S. Byrd, High-fidelity state transfer over an unmodulated linear XY spin chain, Phys. Rev. A 81 (2010) 042313. [6] F. Verstraete, M.A. Martín-Delgado, J.I. Cirac, Diverging entanglement length in gapped quantum spin systems, Phys. Rev. Lett. 92 (2004) 087201. [7] J. Fitzsimons, J. Twamley, Globally controlled quantum wires for perfect qubit transport, mirroring, and computing, Phys. Rev. Lett. 97 (2006) 090502. [8] S. Maniscalco, F. Francica, R.L. Zaffino, N. Lo Gullo, F. Plastina, Protecting entanglement via the quantum Zeno effect, Phys. Rev. Lett. 100 (2008) 090503. [9] J. Ma, Z. Sun, X.G. Wang, Franco Nori, Entanglement dynamics of two qubits in a common bath, Phys. Rev. A. 85 (2012) 062323. [10] Y.H. Ji, Y.M. Liu, Regulation of entanglement and geometric quantum discord of hybrid superconducting qubits for circuit QED, Inter. J. Theor. Phys. 52 (2013) 3220. [11] Y.H. Ji, Y.M. Liu, Investigations into quantum correlation of coupled qubits in a squeezed vacuum reservoir, Chin. Phys. B 22 (2013) 020305. [12] C.E. Lopez, G. Romero, F. Lastra, E. Solano, J.C. Retamal, Sudden birth versus sudden death of entanglement in multipartite systems, Phys. Rev. Lett. 101 (2008) 080503. [13] J.G. Li, J. Zou, B. Shao, Entanglement backflow under the composite effect of two non-markovian reservoirs, Phys. Lett. A 376 (2012) 1020. [14] M. Salerno, V. Popkov, Optimal transport and von Neumann entropy in a Heisenberg XXZ chain out of equilibrium, Phys. Rev. E 87 (2013) 022108. [15] Z.M. Wang, B. Shao, P. Chang, J. Zou, Enhanced entanglement transfer by phase shift control in two parallel spin chains, J. Phys. A 40 (2007) 9067. [16] O. Derzhko, Taras Verkholyak, Taras Krokhmalskii, Helmut Büttner, Dynamic probes of quantum spin chains with the Dzyaloshinskii-Moriya interaction, Phys. Rev. B 73 (2006) 214407. [17] K. Maruyama, T. Iitaka, F. Nori, Enhancement of entanglement transfer in a spin chain by phase-shift control, Phys. Rev. A 75 (2007) 012325. [18] J. Zhang, B. Shao, B.Q. Liu, J. Zou, Q.S. Li, L.A. Wu, Quantum correlation transfer through two parallel XXZ spin chains with phase shift control, Phys. Rev. A. 84 (2011) 012327. [19] M. Brockmann, A. Klümper, Vadim Ohanyan, Exact description of magnetoelectric effect in the spin-1/2 XXZ chain with Dzyaloshinskii – Moriya interaction, Phys. Rev. B 87 (2013) 054407. [20] Z.D. Hu, Q.L. He, J.B. Xu, Quantum communication in spin chains with three-site and Dzyaloshinsky – Moriya interactions, Physica A 391 (2012) 6226. [21] W.K. Wootters, Entanglement of formation of an arbitrary State of two qubits, Phys. Rev. Lett. 80 (1998) 2245. [22] H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett. 88 (2001) 017901. [23] F.F. Fanchini, T. Werlang, C.A. Brasil, L.G.E. Arruda, A.O. Caldeira, Non-Markovian dynamics of quantum discord, Phys. Rev. A 81 (2010) 052107. [24] K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures, Rev. Mod. Phys. 84 (2012) 1655. [25] M. Ali, A.P.R. Ran, G. Alber, Quantum discord for two-qubit X states, Phys. Rev. A 81 (2010) 042105. [26] J.S. Zhang, Ai-X. Chen, Review of quantum discord in bipartite and multipartite systems, Quantum Phys. Lett. 1 (2012) 69.
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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Quantum entanglement transfer through two XXZ spin chains Ying-Hua Ji a,c,∗ , Yong-Mei Liu b a
Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China c Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China b
a r t i c l e
i n f o
Article history: Received 25 April 2014 Accepted 2 June 2015 Available online xxx Keywords: Quantum communication Quantum entanglement Dzyaloshinsky – Moriya interaction Three-site interaction Phase shift
a b s t r a c t We investigated the quantum entanglement through two independent spin chains with phase transfer √ shift. The initial target states 01 + 10 / 2 or its mixed state is shown to be more suitable for quantum correlation transfer in this system. Generating a phase shift by the Aharonov – Casher effect partly improves the amplitude of the entanglement, whereas it is unfavorable for prolonging the survival time of quantum entanglement. We compared and analyzed the effects of spin chains parameters on quantum discord and quantum entanglement. The result indicated that the quantum entanglement is more robust than quantum discord in the two parallel XXZ spin chains. © 2015 Elsevier GmbH. All rights reserved.
PACS: 03.67.Hk 03.65.Ta 75.10.Pq
1. Introduction Transferring quantum information reliably and efficiently is an important task in quantum information processing [1–3]. Most common approaches to this task include methods with an information bus, guided ions, flying photons, a sequence of swapping operations between the neighboring qubits, etc. However, these methods require additional complexity in structures, manipulations, and controls of the interaction between qubits, as well as the repeated conversions between the qubit state and another physical degree of freedom. Spin chains have been proved to be efficient quantum channels for short distance quantum communications and much effort has been devoted to explore efficient schemes to transfer quantum information perfectly or to enhance the quality effectively [4,5]. It has been reported that perfect state transfer through spin networks of arbitrary length can be achieved by preengineered qubit couplings, local measurements on the individual spins [6], global pulses [7], etc. Abundant and reliable information resource can provide the perfect transfer of information with important protection.
∗ Corresponding author at: Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China. Tel.: +86 13755658129. E-mail address: [email protected] (Y.-H. Ji).
Quantum entanglement has been considered as a valuable resource and it plays a central role in various quantum information processes [8]. However, decoherence is the most important limit for quantum computation because its effect is that the quantum superposition decays into statistical mixtures [9,10]. It is known that the coherence and disentanglement are closely connected. Therefore, an increasing interest to study the quantum mechanism is the entanglement dynamics in quantum computation and quantum information. In 2004, it was found that, when the two-level atoms with an initially entangled state interact respectively with a separate vacuum bath, the entanglement between atoms completely disappears after a short time which is much shorter than that of decoherence. This phenomenon was called as the entanglement sudden death (ESD) [11]. The previous results show that ESD is sensitive to the initial conditions of the qubits and the interacting environments. In order to ensure the correct computation of quantum logic and quantum computation, researchers have done many theoretical and experimental studies on the phenomenon of ESD and shown how to avoid ESD and produce the scheme of sudden birth for some cases during the past decade [12]. How to generate and transfer entanglement is a problem of fundamental interest in quantum information processes [13]. In order to overcome the influence of decoherence, one of the tactics proposed by the electrical engineers and researchers is to improve the decoherent time of the qubits by optimizing the system [14].
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Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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Intensive study on the internal and external factors that influence the dynamical behavior of the system is the prerequisite for optimizing the design of quantum system. Recently, there is a growing interest in the study of spin chains with extra couplings, such as Dzyaloshinsky – Moriya interaction, three-site interaction, phase shift, and energy current [15,16]. The results indicate that the teleportation fidelity of thermal entanglement can be enhanced by Dzyaloshinsky – Moriya interaction, and the maximum discord can be enhanced by the phase shift produced by Aharonov – Casher effect [17]. In spite of many works in the area of quantum information transferring, to the best of our knowledge, there are two interesting questions which remain open: (i) What happens to the quantum entanglement transfer in the spin chains? (ii) Which kind of initial quantum state should be prepared so that the ESD can be effectively avoided and then the quantum information can be transferred continuously through two parallel spin chains of arbitrary length? So, in this paper, we present a detailed analysis on quantum entanglement transfer through two parallel XXZ spin chains.
(2) where, ϕ is the phase change between the neighboring spins, which is produced by the Aharonov – Casher effect. ˆ n˛ (˛ = x, y, z; n = 1, 2, . . ., N) is the Pauli matrix for the site n and respectively satisfies the commutation and anti-commutation algebraic relation, ± = ( x ± i y )/2. The first and second terms are the coupling ones. J is the coupling coefficient in x and y directions and Jz is the coupling coefficient in z direction. This model is reduced to the isotropic XX model when Jz =0 and to the isotropic XXX model when Jz = J. The ˛ periodic boundary conditions N+1 = 1˛ are assumed. For the case of the open chain, the phase shift has no effect on the energy spectrum and therefore causes no change in the quantum entanglement transfer. In order to diagonalize the projected Hamiltonian, we first apply the Jordan – Wigner transformations to map spin to onedimensional spinless fermions with creation and annihilation operators as follows [19]: (3)
l=1
n−
=
k−1
1 − 2a+ a l l
H (j) =
(8)
k>0
with
(j)
Ek = J cos (k + ϕ) +
(N − 4)JZ (N − 2)h , − 4 2
other
= 02 . . .0N
(AB)
(j)
(AB)
(AB) (0) = 1−1 (0) ⊗ other (0),
(AB)
(10)
0. The initial general state 1−1 (0) where, other (0) = 0 0ther other of the first spin pair can be written in Bloch representation (AB)
(AB)
(AB)
(AB) 1−1 (0)
1 = 4
(AB) I1−1
+
3(A) b4 1
⊗ I + b5 I ⊗
3(B) 1
+
3
˛(A) b˛ 1
⊗
˛(B) 1
,
˛=1
(11)
(AB)
where, b˛ = Tr 1−1 ( ⊗ )
(AB)
for ˛ = 1, 2, 3 and I1−1 is the
operator in the subspace of the first identity spin pair. b4 = 3(A)
1
⊗I
(AB)
, b5 = Tr 1−1
3(B)
I ⊗ 1
are the compo-
(AB) (t) = U(t)(AB) (0)U + (t),
(12)
where, U(t) = exp(− iHt). Following the procedures presented in Ref. [20], we can obtain the reduced density matrix for spins n and n’ by tracing over all other degrees of freedom nn (t) as
⎛
11
⎜ ⎜ 0
0
0
14
22
23
0
⎝ 0
32
33
0
0
n−n = ⎜ ⎜ (AB)
41
⎞
⎟ ⎟ ⎟. ⎟ 0 ⎠
(13)
44
The diagonal elements of the reduced density matrix (t) for the chain system are: 11 (t) = 1 −
2 1 + b3 + b4 + b5 2 + b4 + b5 fn (t)4 , fn (t) + 2 2
(14)
22 (t) =
2 1 + b3 + b4 + b5 1 + b5 fn (t)4 , fn (t) − 2 2
(15)
33 (t) =
2 1 + b3 + b4 + b5 1 + b4 fn (t)4 , fn (t) − 2 2
(16)
44 (t) = 1 − [11 (t) + 22 (t) + 33 (t)] , an ,
(9)
Assume that initially the first spin pair is prepared in the general (AB) state 1−1 (0) and the others are set in the ferromagnetic ground
(4)
and the nondiagonal elements are
(5)
14 (t) =
l=1
nz = 2a+ n an − 1.
Ek ck+ ck ,
nents of the local Bloch vectors. The time evolution of the total density of the system is given by
n=1
1 − 2a+ a , l l
where, k = 2m/N with −N/2 ≤ m ≤ N/2. Under the one-magnon condition, the Hamiltonian (1) has the following diagonal form
(AB)
1 iϕ x x y y z = J e n n+1 + e−iϕ n n+1 + Jz nz n+1 + 2B nz , 4
n+ = a+ n
(7)
n
Tr 1−1
N
1 ink + ck+ = √ e an , N
(1)
where, H(j) (j = A, B) denotes the Hamiltonian of the jth XXZ spin chain. Considering the effect of the phase shift, the transformed H(j) can be expressed as
n−1
(6)
n
(j)
It is well known that both the Aharonov – Casher effect and the Dzyaloshinsky – Moriya interaction can generate the phase shift in a spin chain, and this phase will manipulate the quantum information transfer. In this paper, the model we consider consists of two parallel XXZ spin chains with the effect of phase shift. Each Heisenberg chain contains N qubits interacting ferromagnetically with their nearest neighbors. The total Hamiltonian of the system can be written as (kB = = 1)[18]
H (j)
1 −ink ck = √ e an , N
state 0
2. Parallel XXZ spin chains with phase shift
H = H A ⊗ I (B) + I (A) ⊗ H (B) ,
Subsequently, the Fourier transforms of the fermionic operators are described by
b3 − b4 2 fn (t)e2iE0 t , 2
(17)
(18)
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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2 b3 + b4 fn (t) , 2
23 (t) = (j) fn (t)
(19)
(20)
k
(A)
∗ (t). In above equations, we have set f (t) = f and jk (t) = kj n n (t) = (B)
fn (t) and E0 = − N/4. Then we can investigate the quantum correlation between spins n and n’ analytically. 3. Measurement of quantum entanglement Characterization of entanglement is one of the key features related to quantum information theory. From the point of view of a possible application, it is important not only to determine whether a given state is entangled, but also to quantify the degree of entanglement. Among several such quantities, the concurrence introduced by Bennet et al. is often used for this purpose. The concurrence C in a bipartite state is defined as [21]
1 −
2 −
3 −
4 },
(21)
where, j are the eigenvalues of the non-Hermitian W = (t)(t) ¯ arranged in descending order 1 ≥ 2 ≥ 3 ≥ 4 . Spin-flip operator (t) ¯ is defined as (t) ¯ =
(1)
(2)
y ⊗ y
∗ (t) y ⊗ y (1)
(2)
.
The asterisk *(t) is the complex conjugate of density matrix (AB) (t). For this n−n (t), the concurrence will be
C(t) = 2 max{0, 23 (t) − −
11 (t)44 (t), 14 (t)
22 (t)33 (t)},
(22)
which is obviously dependent on the initial conditions. It is indisputable that certain amplitude and duration are necessary for quantum communication and quantum computation. Therefore, this section will focus on the evolution characteristics of the entanglement dynamics of the spin in each position and accordingly determine if the quantum information can be transferred efficiently. We here consider the X-states, which may arise in a wide variety of physical situations, as the target state to be transferred to investigate the transfer properties of quantum entanglement. X-states are also encountered as eigenstates in all the systems with odd-even symmetry like in the Ising and the spin chain models. X-states are defined as:
r, = r (ϕ)
˚ r, = r ˚()
(ϕ) +
˚() +
2
Cn−n = fn (t)
r 1 − cos 2 − 1 + r cos 2
1 (j) = exp ik (n − 1) − iEk t , N
C(t) = max{0,
3
1−r I4 , 4
(23a)
1−r I4 . 4
(23b)
− 1 − r + 4r cos2
2
˚ = fn (t) Cn−n
fn (t)2 ,
r 1 − cos 2
(24)
2 4 − (1 − r) 2 − 2 fn (t) + (1 − r) fn (t) .
(25)
These equations are quite general since their forms are independent on the particular sites of the spin chains, but only on the Hamiltonian model of Eq. (1) and on the chosen initial state. Once the parallel XXZ spin chains structure are specified, the explicit form of fn (t) is obtained and in turns the explicit time-dependence of concurrence is determined. 4. Quantum entanglement transfer In this section we investigate the transfer properties of two entangled qubits transferring through two identical uncoupled chains A and B of length N, each of which by the Hamil is modeled tonian (2). For the initial target state r, , we plot the quantum entanglement that transfers from the first sites of the dual chains to the (n,n) th sites as a function of rescaled time Jt in Fig. 1 and Fig. 2. Figs. 1 and 2 clearly show that the death and rebirth happens to concurrence with the variation of Jt in the target state r, . We also find that for smaller N, quantum entanglement can be transferred from the first spin pair with and without the phase shift to the nth any other spin chain for various (N,n). There are some common features for the concurrence. First of all, for a fixed n, we find that value of the concurrence of nth spin chain decreases with increasing N. Second, in general, entanglement death occurs more easily and entanglement rebirth more difficultly as the transmission distance increases (namely, n increases). Finally, based upon huge numerical computations, we find that when N is fixed, the transmission properties are not obviously improved by manipulating the coupling coefficients and external magnetic intensity. Comparing Figs. 1 and 2, however, the quantum entanglement can be enhanced by a nonzero phase shift for most time, which agrees well with the results in the reference. Particularly, Figs. 1–4 reveal that the phase shift just partly improves the amplitude of
In Eq. (23), r is the purity of the initial states, which ranges from 0 for maximally mixed states to 1 for pure states, I4 is the 4 × 4 identity matrix and
() = cos 00 + sin 11 , ˚() = cos 01 + sin 10 ,
are the Bell-like pure state, and the parameter is sometimes called as the degree of entanglement. The explicit expressions at time t are given for of concurrences two initial states r, and ˚ r, of Eq. (22), respectively, by
Fig. 1. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaledtimeJt without phase shift. The initial state of
the first spin pair is prepared in
r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
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Fig. 5. The quantum entanglement that transfers from sites (1, 1) to sites (N − 1, N − 1) as a function of rescaled time Jt with phase shift ϕ = /3. When the chain length: (a) N = 15, (b) 31. The initial state of the first spin pair is prepared in Bell state with JZ = 0.1J, B = J.
Fig. 2. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaled time Jt with phase shift ϕ = /3. The initial state of the first spin pair is prepared in
r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
entanglement and does the maintenance of entanglement no good. In addition, the transfer quality can be enhanced by increasing the purity of the initial states. We also examine the case of the initial target state ˚ r, as shown in Fig. 3 and Fig. 4. By comparing Figs. 3 and 4 with Figs. 1 and 2, it is interesting to find that the effects of phase are very similar for state shift on the quantum entanglement r, and state ˚ r, . The transfer quality for the r,
case is always lower than that for the ˚ r,
case. If the ini-
tial target state is ˚ r, , in the evolution of entanglement, once ESD occurs to the concurrence, almost simultaneously the entanglement rebirth appears, which is beneficial to the information transmission. When the spin chains are larger, as shown in Fig. 5, N = 31, excellent entanglement quality can still be obtained inposition (30,30). Inversely, if the initial target state is r, , the entanglement in the same position is quite small and rebirth is difficult for the entanglement, the evolution quality is poor. Further numerical computations indicate that using the initial target state ˚ r, is more suitable for quantum correlation transfer in this system. Moreover, the transmission quality can be improved obviously via enhancing the purity of the initial state. 5. Quantum discord transfer
Fig. 3. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaledtimeJt without phase shift. The initial state of
the first spin pair is prepared in ˚ r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
Fig. 4. The quantum entanglement that transfers from sites (1,1) to sites (n,n) (here n = 2,3,4,5) as a function of rescaled time Jt with phase shift ϕ = /3. The initial state of the first spin pair is prepared in ˚ r,
with JZ = 0.1J, B = J, r = 0.9 and = /4.
It has been realized that entanglement represents only a special kind of correlations and there exist other nonclassical correlations such as quantum discord first introduced in Ref. [22]. In many models, it has been reported both theoretically [23] and experimentally [24] that the quantum discord is non-zero even for unentangled states and more robust than the entanglement against sudden death [25], which indicates that quantum information processing based on quantum discord may be more robust than that based on entanglement. However, the quantum discord of a general bipartite state is not always larger than the entanglement, which means that the quantum discord is not simply the sum of entanglement and some other non-classical correlation. The relation between entanglement and quantum discord is complicated for general states. For the pure entangled states, quantum discord coincides with the entropy of entanglement. However, these two measurements disagree on the quantum correlations of a mixed state because the quantum discord may be nonzero for some separable states. The entanglement is more robust than quantum discord in some systems. Therefore, choosing entanglement or discord in a determined quantum system as the information resource remains an important and interesting question. In this section, we simply investigate quantum discord transfer through two identical uncoupled chains A and B of length N, each of which is modeled by the Hamiltonian (1). Currently, how to measure quantum discord is still a hot topic. In order to study the relation between entanglement and quantum discord, we select geometric measure of quantum discord (GMQD)
Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008
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are more suitable for quantum correlation transfer in this system. We compare and analyze the effects of spin chain parameters on the quantum discord and quantum entanglement. The result indicates that the quantum entanglement is more robust than quantum discord in the two parallel XXZ spin chains. Acknowledgment
Fig. 6. The quantum correlation that transfers from sites (1,1) to sites (N − 1, N − 1) chain length (a) as a function of rescaled time Jt with phase shift ϕ = /3. Whenthe /√2. N = 11, (b) 21. The initial state of the first spin pair is prepared in 01 + 10
as the physical quantity for measuring quantum correlation. The GMQD, a measurement of quantum correlations in a quantum system, is defined as [26]
2
DA () = min − , g
∈˝0
(26)
where, the minimum is larger than all possible classical states of the form p1 ˚1 ˚1 ⊗ 1 + p2 ˚2 ˚2 ⊗ 2 with p1 + p2 = 1.
˚1 and ˚2 are two orthogonal bases of the subsystem A. 1 and 2 2 are two density matrices of the subsystem B. Here, − =
Tr( − )2 represents the square norm of the Hilbert – Schmidt space. In Fig. 6, we plot the dynamics of quantum and√entan discord glement for the for the initial target states 01 + 10 / 2 as a function of the scaled time Jt. Fig. 6 shows that the common feature of the dynamics for both of them is a decrease with increasing N. However, the entanglement has stronger robustness than quantum discord. The characteristics of quantum discord transfer are qualitatively different from that of entanglement. We clearly find that: (1) As a whole, the quantum discord is very weak and easily tends to zero in a short time, which is of no practical use. (2) Entanglement survives longer for a fixed N, and the quantum discord death is faster on the contrary. (3) The particularly important difference is √ that: for the initial target states 01 + 10 / 2, it is difficult for the quantum discord to rebirth after the death appears, and that more greater N, more difficult to rebirth. By contrast, rebirth can occur fast to the quantum entanglement as addressed above, which verifies again that the quantum entanglement is qualitatively different from quantum discord. 6. Conclusions We have investigated how the length N or site n of two parallel XXZ spin chains and the Aharonov – Casher effect influence the quantum entanglement when the system is initially in X-state. We find that the time evolution of quantum entanglement is oscillating, and collapse and revival randomly occur. With the increasing of N, the quantum entanglement was quite small and the duration was too short to be used as the quantum information resource. Numerical simulations indicate that the phase shift merely partly improves the amplitude of entanglement and is disadvantageous to We also find prolong the survival time ofquantum entanglement. √ that the initial target states 01 + 10 / 2 or in its mixed state
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Please cite this article in press as: Y.-H. Ji, Y.-M. Liu, Quantum entanglement transfer through two XXZ spin chains, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.008