Teleportation via thermally entangled state of a two-qubit Heisenberg XXX chain

Optik 127 (2016) 8475–8478

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Teleportation via thermally entangled state of a two-qubit Heisenberg XXX chain夽 Cheng Li a,b,∗ , Nian-shun Zhao a a

School of Information Engineering, Huangshan University, Huangshan 245000, China Key Laboratory of Opto-electronic Information Acquisition and Manipulation of Ministry of Education, Anhui University, Hefei 230039, China b

a r t i c l e

i n f o

Article history: Received 18 April 2016 Accepted 8 June 2016 PACS: 03.65.Yz 42.50.-p Keywords: Quantum entanglement Thermal entanglement Teleportation Fidelity Quantum phase transition

a b s t r a c t In this paper, we consider the 1D two-qubit isotropic antiferromagnetic Heisenberg XXX model in an external magnetic field B. Firstly, we pay our attention to study the properties of thermal entanglement in XXX model, and in comparison with XX model, we found that the thermal entanglement could be easily obtained and quantum phase transition could not easily occur in XXX model. Finally, we calculate the fidelity of teleportation of one-qubit pure state via thermally entangled state of a two-qubit Heisenberg XXX chain, in the same way, we compare XXX model with XX model, we found that teleportation via thermally entangled state of a two-qubit Heisenberg XXX chain may be better. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction Quantum entanglement is one of the most striking features of quantum mechanics. Because of its unique properties [1,2], quantum entanglement plays an important role in quantum information processing, such an quantum teleportation [3], superdense coding [4], etc. In particular the presence of entanglement in condensed-matter systems at finite temperatures has been widely investigated, the state of a typical condensed-matter system at thermal equilibrium (temperature T) is  = e−ˇH /Z. Recently, the concept of thermal entanglement was introduced and investigated within one dimensional isortopic Heisenberg model [5,6]. Wang considered the two-qubit isotropic antiferromagnetic XY model in an external magnetic field B. He found that there is a critical temperature Tcritical = arcsinJ h(1) ≈ 1.13459J which is independent on the magnetic field B. Beyond Tcritical the thermal entanglement vanishes [7]. Ye found that quantum teleportation, using the thermally entangled state of a two-qubit Heisenberg XX chain as a resource, with fidelity better than any classical communication protocol is possible. However, a thermal state with a greater amount of thermal entanglement does not yield this. In fact, it depends on the amount of mixing between the separable state and maximally entangled state in the spectra of the twoqubit Heisenberg XX model [7]. The Heisenberg interaction has been used to implement quantum computer [8,9], by suitable

夽 Foundation item: supported by the Foundation of Key Laboratory of Opto-electronic Information Acquisition and Manipulation of Ministry of Educationthe (OEIAM201413), the Natural Science Foundation of the Education Department of Anhui Province (KJHS2015B07). ∗ Corresponding author at: School of Information Engineering, Huangshan University, Huangshan 245000, China. E-mail address: [email protected] (C. Li). http://dx.doi.org/10.1016/j.ijleo.2016.06.037 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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coding, the Heisenberg interaction alone can be used for quantum computation [10,11]. So we think it is significant to study all kinds of Heisenberg model. In this paper, we consider the 1D two-qubit isotropic antiferromagnetic Heisenberg XXX model in an external magnetic field B. In Section 2, we pay our attention to study the properties of thermal entanglement in XXX model. In Section 3, we calculate the fidelity of teleportation of one-qubit pure state. 2. Thermal entanglement in XXX model We consider the 1D two-qubit isotropic antiferromagnetic Heisenberg XXX model in an external magnetic field B, the Hamiltonian of this system can be written as H=

 J 1  1  1 x ⊗ x2 + y1 ⊗ y2 + z1 ⊗ z2 B z + z2 + 2 2

(1)

xi , yi , zi are Pauli operators for the ith qubit. The eigenvalues and eigenvectors of H are obtained as H|00 >= (B − 2J )|00 > H|11 >= (−B − 2J )|11 > H| + >=

3J | + 2

>

H| − >= − 2J | − > where | ± >=

√1 (|01 2

> ±|10 >)

The density matrix of the state at the equilibrium temperature T can be expressed as  =  thermal entangled  exp(−H/kT )/Z, where Z = tr exp(−H/kT ) is the partition function and k is the Boltzmann’s constant. k is set to 1.The entanglement in the state is called thermal entanglement [12]. In the standard basis, {|00 > ,|01 > ,|10 > ,|11 > }, the density matrix  is expressed as

⎛

−B−

⎜ ⎜e T ⎜ ⎜ ⎜ ⎜ 0 1⎜ = ⎜ Z⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎝

⎞

J 2

0 1 2 1 2



e

−

J 2T

J 2



3J − e 2T



1 2 1 2



−B−

J 2

J

0

3J

J



e− 2T − e 2T

 e

0

0 B−

3J

J

e− 2T + e 2T



⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ J ⎟ ⎠ B−

0

−

J 2T

0

3J + e 2T



e

T

(2)

2

3J

where Z = e T + e T + e− 2T + e 2T , to quantify the amount of entanglement associated with , we consider the concurrence [13,14]. C = max{1 − 2 − 3 − 4 , 0} where the quantities 1 ≥ 2 ≥ 3 ≥ 4 are the square roots of the eigenvalues of the operator R = (y ⊗ y )∗ (y ⊗ y ). After some straightforward calculation, the concurrence is 3J

J

e 2T − 3e− 2T

C () = max{

B−

e

T

J 2

+e

−B− T

J 2

+e

, 0} J

− 2T

+e

(3)

3J 2T

We found that there is a critical temperature Tcritical =

2J ln(3)

≈ 1.82048J

(4)

Which is independent on the magnetic field B, the entanglement vanisher for T ≥ Tcritical . In comparison with XX model, the critical temperature is greater in XXX mode, this indicates that the thermal entanglement could be easily obtained in XXX model. As seen from Fig. 1, for B = 0, the maximal entanglement is at T = 0, the ground state with eigenvalue − 2J | − >. As T increases, the concurrence decreases. For a high value of B (B = 2.2), there is no entanglement at T = 0, however, as T increases, the entanglement increases. From Fig. 2, we see that for small temperature, when we increase magnetic field, we will find the evidence of phase transition. Now we calculate the limit T → 0 on the concurrence (3), we obtain Lim C = 1forB < 2J, T →0

C. Li, N.-s. Zhao / Optik 127 (2016) 8475–8478

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Fig. 1. The concurrences versus temperature for different magnetic fields. The parameter J is set to one.

Fig. 2. The concurrences versus magnetic field B for different temperatures. The parameter J is set to one.

Lim C =

T →0

1 forB 2

= 2J,

Lim C = 0forB > 2J,

(5)

T →0

So at T = 0, as B crosses the critical value 2J, the entanglement vanishes. This particular point T = 0,B = 2J, at which entanglement becomes a nonanalytic function of B, is the point of quantum phase transition [15]. In comparison with XX model, quantum phase transition could not easily occur in XXX model. 3. Teleportation via thermally entangled state Now we look at the standard teleportation protocol P0 , using the above two-qubit mixed state  as a resource. We consider as input a qubit in an arbitrary pure state | >= cos 2 |0 > +ei sin 2 |1 >(0 ≤  ≤ , 0 ≤  ≤ 2), The output state is given by [16]. P0 ()|

><

|=

3

tr(E j )j |

><

|j

(6)

j=0

where E 0 = | − ><  − |, E 1 = |˚− >< ˚− |, E 2 = |˚+ >< ˚+ |, E 3 = | + ><  + |, and |˚± >=

√1 (|00 2

> ±|11 >). It fol-

lows that <

|[ P0 ()|

|]|

><

>=

3J 3J J J J − − 1 1 − B sin2 (−e 2T + e 2T ) + sin2 e 2T cosh + (e 2T + e 2T )(3 + cos 2) 2 T 4 J J B− −B − J 3J 2 2 − T e T +e + e 2T + e 2T

(7)

And averaging over all possible input states we obtain the fidelity of the teleportation J

3J

J

e− 2T + 3e 2T + 2e− 2T cosh 3(e

J B− 2 T

+e

J −B− 2 T

+e

J

− 2T

+e

B T 3J 2T

(8) )

This is the maximal fidelity achievable from  in the standard teleportation scheme P0 [17,18].

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Table 1 m is function of both J and B = J, 0 < < J. The C gives the amount of thermal entanglement below which does not yield teleportation fidelity better than Tcritical any classical commutation protocol [19]. m Tcritical

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1.818811J 1.813794J 1.805388J 1.793522J 1.778097J 1.758973J 1.735970J 1.708856J 1.677333J 1.641018J 1.599413J 1.551864J 1.497485J 1.435042J 1.362730J 1.277738J 1.175252J 1.045647J 0.862988J

C 0.000503298 0.00202028 0.00457038 0.00818824 0.0129229 0.0188431 0.0260382 0.0346235 0.0447479 0.0566034 0.0704399 0.0865868 0.105489 0.127762 0.154298 0.186452 0.226457 0.226457 0.352225

In order to transmit | > with fidelity better than any classical communication protocol, we require (8) to be strictly greater than 2/3.In other words, we require 2J

B

B

e T > e T + e− T + 1

(9)

m And hence B < 2J. The ‘critical’ temperature Tcritical beyond which the performance is worst than what classical communication protocol can offer, is clearly dependent on the magnetic field B. As seen from Table 1, T m critical decreases with increasing B

and they are all less than Tcritical = ln2J3 ≈ 1.82048J. By comparison with XX model, when B in XXX model is equal to B in XX m m in XXX model is greater than Tcritical in XX model, we found teleportation via thermally entangled state may model, Tcritical be better in XXX model.

4. Conclusion In this paper, we study the properties of the 1D two-qubit isotropic antiferromagnetic Heisenberg XXX model in an external magnetic field B, and teleportation via thermally entangled state in the XXX model, In comparison with XX model, we found the thermal entanglement could be easily obtained, quantum phase transition could not easily occur and teleportation via thermally entangled state may be better in XXX model. From the above-mentioned analysis, we found every kind of Heisenberg model has itself priority, so we should select appropriate model according to our purpose. References [1] A. Einstein, B. podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete, Phys. Rev. 47 (1935) 777. [2] Wei Chen, Zheng-Yuan Xue, Z.D. Wang, Shen Rui, All-electrically reading out and initializing topological qubits with quantum dots, Commun. Theor. Phys. 23 (2014) 1056–1674. [3] C.H. Bennett, G. Brassard, C. Crepeau, et al., Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70 (1993) 1895. [4] C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without Bell’s theorem, Phys. Rev. Lett. 68 (1992) 557. [5] M.C. Arnesen, S. Bose, V. Vedral, Natural thermal and magnetic entanglement in 1D heisenberg model, Phys. Rev. Lett. 87, 017901. [6] M.A. Nielsen, Quantum information theory, Ph.D. thesis, The University of New Mexico, eprint quantph/0011036. [7] X.G. Wang, Entanglement in the quantum heisenberg XY model, Phys. Rev. A 64 (2001) 012313. [8] Y. Yeo, Teleportation via thermally entangled state of a two-qubit Heisenberg XX chain, Phys. Rev. A 66, 062312. [9] D. Loss, D.P. Divincenzo, Quantum computation with quantum dots, Phys. Rev. A 57 (1998) 120. [10] G. Burkard, D. Loss, D.P. Divincenzo, Coupled quantum dots as quantum gates, Phys. Rev. B 59 (1999) 2070. [11] D.A. Lidar, D. Bacon, K.B. Whaley, Concatenating decoherence-free subspaces with quantum error correcting codes, Phys. Rev. Lett. 82 (1999) 4556. [12] D.P. Divincenzo, D. Bacon, J. Kempe, et al., Universal quantum computation with the exchange interaction, Nature 408 (2000) 339. [13] S. Hill, W.K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78 (1997) 5022. [14] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245. [15] S. Sachdev, Quantum Phase Transitions, Cambridge, Cambridge University Press, 1999. [16] G. Bowen, S. Bose, Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity, Phys. Rev. Lett. 87 (2001) 267901. [17] Chuan-Jia Shan, Shuai Cao, Zheng-Yuan Xue, Shi-Liang Zhu, Anomalous temperature effects of the entanglement of two coupled qubits in independent Environments, Commun. Theor. Phys. 29 (2012) 0256–307X. [18] M. Horodecki, P. Horodecki, R. Horodecki, General teleportation channel,singlet fraction, and quasidistillation, Phys. Rev. A 60 (1999) 1888. [19] I.P. Degiovanni, M. Genovese, V. Schettini, M. Bondani, A. Andreoni, Monitoring the quantum-classical transition in thermally seeded parametric down-conversion by intensity measurements, Phys. Rev. A 79 (2009) 063836.

Cheng Li (1979)—Senior lecturer, engaged in the studies on quantum optics and optic fiber communication.