(Received 22 October 2009; revised manuscript received 30 December 2010)

Transcription

1 Chin. Phys. B Vol. 19 No ) Teleportation and thermal entanglement in two-qubit Heisenberg XY Z spin chain with the Dyaloshinski Moriya interaction and the inhomogeneous magnetic field Gao Dan 高丹 ) a) Zhao Zhen-Shuang 赵振双 ) a) Zhu Ai-Dong 朱爱东 ) a) Wang Hong-Fu 王洪福 ) b) Shao Xiao-Qiang 邵晓强 ) b) and Zhang Shou 张寿 ) a) a) Department of Physics College of Science Yanbian University Yanji China b) Center for the Condensed-Matter Science and Technology Department of Physics Harbin Institute of Technology Harbin China Received October 009; revised manuscript received 30 December 010) This paper studies the average fidelity of teleportation and thermal entanglement for a two-qubit Heisenberg XY Z chain in the presence of both an inhomogeneous magnetic field and a Dyaloshinski Moriya interaction. It shows that for a fixed D the increase of b will broaden the critical temperature at the cost of decreasing the thermal entanglement. And it can modulate the inhomogeneous magnetic field and the Dyaloshinski Moriya interaction for the average fidelity of teleportation to be optimal. Keywords: teleportation thermal entanglement Heisenberg XY Z spin chain PACC: H In recent years entanglement has been extensively studied because it has the fascinating features of quantum mechanics and it provides a fundamental resource in quantum information processing such as quantum teleportation 1] quantum dense coding 34] quantum cryptographic key distribution 56] quantum computation 78] and so on. As one knows that quantum teleportation process allows the two distant parties the sender Alice and the receiver Bob to utilise the nonlocal correlations of the quantum channel initially shared by Einstin Podolsky Rosen EPR) 9] pair to teleport an unknown quantum state φ = a + b Alice makes a Bell-basis measurement on her EPR particle and the unknown quantum system then Bob can reconstruct the state φ with a local unitary operation on his EPR particle according to the classical information published by Alice. Generally for a perfect quantum teleportation scheme the precondition is to find the quantum channel. Recently the teleportation with some solid systems such as quantum chain becomes an attractive emerging field 10 1] A quantum chain is usually referred as a spin chain which is a one-dimensional array of qubits coupled permanently by mutual interaction. Lately great efforts have been taken to study the teleportation in the one-dimensional Heisenberg chain 13 ] as it can be considered as an important source of entanglement. Yeo has studied the teleportation in thermal entangled state of a two-qubit Heisenberg XX chain 3] Zhang studied thermal entanglement and teleportation in a twoqubit Heisenberg chain with Dyaloshinski Moriya DM) interaction. 4] Very recently the Heisenberg model including Ising model 5] XY model 6] XXX model 7] XXZ model 8] and XY Z model 9] have also been widely studied. In this paper we study the quantum teleportation and thermal entanglement via a two-qubit Heisenberg XY Z spin chain by introducing the Z-component DM interaction and an inhomogeneous magnetic field along the Z-component in the presence of an external uniform magnetic field. By modulating the corresponding parameters the thermal entanglement and the average fidelity can be efficiently controlled. Project supported by the National Natural Science Foundation of China Grant No ). Corresponding author. shang@ybu.edu.cn c 010 Chinese Physical Society and IOP Publishing Ltd The model Hamiltonian of our system can be de

2 Chin. Phys. B Vol. 19 No ) scribed by H = J x σ x 1 σ x + J y σ y 1 σy + J σ 1σ +D σ x 1 σ y σx σ y 1 ) +B + b )σ 1 + B b )σ 1) J i i = x y ) are the real coupling coefficients with J i > 0 corresponding to the antiferromagnetic case and J i < 0 corresponding to the ferromagnetic case D is the Z-component DM coupling parameter B uniform external magnetic field) and b nonuniform external magnetic field) are the Z-component magnetic field parameters and σ i are the Pauli matrices. Under the standard bases { } the matrix form of Hamiltonian Eq. 1) can be expressed as J + B 0 0 J x J y 0 J b ) J x + J y + D i 0 H =. ) 0 J x + J y D i J + b ) 0 J x J y 0 0 J B Without loss of generality after some straightforward calculations the eigenvectors of this channel Hamiltonian is Ψ 1 = sin θ cos θ 1 11 Ψ 34 = sin θ χ cos θ ) with the corresponding eigenvalues E 1 = J ± E 34 = J ± ω 4) = 4B + J x J y ) ω = 4b + 4D + J x + J y ) χ = J x + J y D i Jx + J y ) + 4D θ 1 = arctan J x J y ± B Jx + J y ) + 4D θ 34 = arctan. ±ω b The state of the system at thermal equilibrium thermal state) is ρt ) = 1/Z)e βh) Z = Tre βh) ] is the partition function of the system H is the Hamiltonian of the system β = 1/k B T T is the temperature and k B is the Boltman constant which is taken to be 1 for simplicity. Therefore the density matrix of the system discussed above can be written as U δ ρt ) = 1 0 U U 3 0 5) Z 0 U 4 U 5 0 δ 0 0 U 6 U 1 = e βj coshβ ) B ] sinhβ ) U = e βj coshβω ) b ] sinhβω ) ω U 3 = χe βj 4D + J x + J y ) sinhβω ) ω 4D U 4 = χ e βj + J x + J y ) sinhβω ) ω U 5 = e βj coshβω ) + b ] sinhβω ) ω U 6 = e βj coshβ ) + B ] sinhβ ) δ = e βj J x J y sinhβ ) the partition function Z is given by Z = e βj coshβ ) + e βj coshβ )]. We now study the quantum teleportation through a two-qubit Heisenberg XY Z chain by using the standard teleportation protocol. 30] Assume that the input state is an arbitrary two-qubit pure state such as Ψ in = cos θ 10 + ei ϕ sin θ 01 0 θ π and 0 ϕ π). 6) Then the output state is given by 31] ρ out = ij=0xy P ij σ i σ j )ρ in σ i σ j ) 7) σ 0 is the identity matrix and σ i i = x y ) are the Pauli matrices ρ in is the density matrix of the input state P ij = Tr E i ρt ) ] TrE j ρt )] ij P ij =

3 Chin. Phys. B Vol. 19 No ) and E 0 = Ψ Ψ E 1 = Φ Φ E = Φ + Φ + E 3 = Ψ + Ψ + Φ ± = 00 ± 11 )/ Ψ ± = 01 ± 10 )/. After the calculations the output density matrix can be expressed as a 0 0 ω ρ out = 1 Z 0 b ε 0 8) 0 ε c 0 ω 0 0 d a = U 1 + U 6 )U + U 5 ) b = U + U 5 ) sin θ ) + U 1 + U 6 ) cos θ ) c = U 1 + U 6 ) sin θ ) + U + U 5 ) cos θ ) d = U 1 + U 6 )U + U 5 ) ω = U 1 + U 6 )U + U 5 ) sin θe i ϕ + e i ϕ ) ω = U 1 + U 6 )U + U 5 ) sin θe i ϕ + e i ϕ ) ε = 1 U 3 + U 4 ) e i ϕ sin θ + δ e i ϕ sin θ ε = 1 U 3 + U 4 ) e i ϕ sin θ + δ e i ϕ sin θ. In order to describe the output quantum entanglement we use Wooters concurrence 3] which varies from C = 0 for a separable state to C = 1 for a maximally entangled state. Since the density matrix in this paper only contains non-ero elements along the main diagonal and anti-diagonal i.e. in an X formation) the concurrence can be calculated as 33] Cρ out ) = max { 0 ϵ ad ω } bc. 9) Figure 1 demonstrates the C out versus temperature T for different inhomogeneous magnetic field b when the input state is a maximally entangled state. We can see that the C out first maintains a stable value at a short range of temperature T then it decreases monotonically with the increase of T. Let To k k = ) be the intersection points for arbitrary two curves shown in Fig. 1. At these points the C out values corresponding to different b are equal. It can be easily seen that when T < To k the C out decreases with increasing b. While when T > To k the C out increases with increasing b meanwhile we obtain a higher critical temperature. Therefore the inhomogeneous magnetic field is beneficial to broaden the critical temperature at the cost of decreasing the concurrence. Fig. 1. Teleported thermal concurrence C out as a function of T for different inhomogeneous magnetic field b with the coupling constant J x = 0.95 J y = 0.05 J = 0. for a finite value of uniform magnetic field B = 0.5 and the DM interaction D = 0.5. The quality of the teleported state ρ out is characterised by the fidelity F ρ in ρ out ) defined by 34] F ρ in ρ out ) = Tr ρ in ) 1/ ρ out ρ in ) )] 1/. 10) For F ρ in ρ out ) = 0 it means that the information is completely destroyed in the transmission process and the teleportation fails. While for F ρ in ρ out ) = 1 it means that the final state is identical to the initial state. In common situation 0 < F < 1 it results in distortions to the quantum information at a certain extent after being transmitted. After a straightforward calculation the fidelity is given by ) F ρ in ρ out ) = U + U 5 ) +4 sin θ θ cos ] +δ cosϕ) sin 4 θ + θ cos4 1 U 3 + U 4 ) +U 1 + U 6 ) sin θ cos θ. 11) As is well known that the fidelity can be regarded as a useful indicator of teleportation quality. When the input state is a pure state the efficiency of quantum communication is characterised by the average fidelity. The average fidelity F A of teleportation can be formulated as 35] F A = π dϕ π F sin θdθ ) 4π Through some straightforward algebra we can obtain the average fidelity as

4 F A = Chin. Phys. B Vol. 19 No ) ] 4 e βj cosh βω ) + e βj cosh β ) + e βj J x+j y ) ω sinh βω ) 3Z. 13) In Fig. the average fidelity F A is plotted as a function of the temperature T under different inhomogeneous magnetic field b. From the figure we can see that the average fidelity F A first maintains a fixed value at a short range of temperature T then it decreases monotonically with the increase of T at last T these four lines superpose. Let T i o i = ) be the intersection points for arbitrary two curves shown in Fig.. When T < T i o the F A decreases with increasing b when T > T i o the F A increases with increasing b. Meanwhile increasing b can broaden the region F A is larger than /3 for the classical case. limited value of fidelity /3 for classical communication. Therefore in order to illustrate the superiority of quantum communication one should take D less than 1.5 in the presence of the inhomogeneous magnetic field. Fig. 3. The average fidelity F A as a function of the DM interaction with the different inhomogeneous magnetic field b at a finite temperature T = 0.1. We work in units D and b are dimensionless. Fig.. The average fidelity F A as a function of the temperature T with different inhomogeneous magnetic field b we set J x = 0.95 J y = 0.05 J = 0. B = 0.5 D = 0.5. In Fig. 3 we plot the average fidelity F A as a function of the DM interaction D with different values of the inhomogeneous magnetic field b at a finite temperature T = 0.1 we set J x = 0.95 J y = 0.05 J = 0. and B = 0.5. From this figure we can see clearly that for small b b = 0) we can get the maximum value of the fidelity F A by increasing the D. Then the average fidelity F A tends to the limited value of fidelity /3 for classical communication following the further increase of D. While for relatively large value of b b = 0.6) the average fidelity F A decreases monotonically with the increase of D to the In conclusion we have studied the quantum teleportation via a two-qubit Heisenberg XY Z chain under the DM coupling parameter and the inhomogeneous magnetic field. Our study shows that both the concurrence and the fidelity decrease with the increase of T and we know that the inhomogeneous magnetic field not only increases the output entanglement when T > T k o but also enhances the critical temperature. We can see that the average fidelity increases with the increase of the inhomogeneous magnetic field when the DM coupling parameter is set to ero. The influence of D on the model resembles the effects of b when b is modulative. In addition the effects of the inhomogeneous magnetic field on the average fidelity and output entanglement are studied for a fixed D. The inhomogeneous magnetic field not only can improve the critical temperature but also can enhance the average fidelity. References 1] Bennett C Brassard G Crepeau C Josa R Peres A and Wootters W 1993 Phys. Rev. Lett ] Cao Z L and Li D C 008 Chin. Phys. B ] Bennett C and Wiesner S 199 Phys. Rev. Lett ] He J Ye L and Ni Z X 008 Chin. Phys. B ] Ekert A K 1991 Phys. Rev. Lett ] Wen H Han Z F Guo G C and Hong P L 009 Chin. Phys. B

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