量子Heisenberg模型热纠缠性质的研究
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摘要
在20世纪末,量子力学拓展了新的研究领域——量子信息,其中量子纠缠是研究的热点。量子纠缠是存在于多体量子系统中的一种奇妙现象,由于纠缠的非局域特征它已被作为一种具有潜存价值的资源在量子计算、量子信息处理等领域发挥着关键作用。通过严格计算系统自旋间的量子纠缠,研究了Heisenberg XY模型和XXZ模型的纠缠问题,得到了一些有意义的结果。其主要内容如下：
在磁场B存在的情况下研究了具有Dzyaloshinski-Moriya (DM)相互：作用(D)的Heisenberg XY模型,讨论了几种物理量对纠缠的影响。发现在低温时磁场B对纠缠值与纠缠特性都有一定的影响,并且还发现根据各向异性参数γ的不同从而影响临界温度的大小。通过研究发现DM相互作用不影响纠缠回升的过程,它只影响临界磁场Bc区域并且调节临界磁场Bc的大小。当J增加到一定值时,无论B和D的值多大,纠缠的最大值都是不变的。
利用Negativity的方法在Heisenberg XXZ模型中计算了具有DM相互作用的自旋间的热纠缠,分析了温度、DM相互作用、各向异性参数、耦合常数和自旋大小对纠缠的影响。通过计算两粒子之间的纠缠,发现铁磁和反铁磁情况下的纠缠情况不同：反铁磁情况下的纠缠值一般大于铁磁情况下的纠缠值。住D相对较小时,纠缠在铁磁和反铁磁情况下的值是不同的。而随着D的增加,纠缠最后会达到相同的值。在DM相互作用下,两个费米子系统的纠缠最大值大于混合系统的纠缠最大值,由此得出,DM相互作用对两个费米子系统的影响比较明显。混合系统在一个相对较小的D值就会达到纠缠最大值,然而两个费米子系统需要较强的DM相互作用才能达到纠缠最大值。
进一步研究了自旋为1的具有DM相互作用的Heisenberg XXX模型和XXZ模型的基态和热纠缠,将结果和(1/2,1/2),(1/2,1)系统的结果比较,发现纠缠值在(1/2,1/2)系统最大,在(1/2,1)系统最小,这意味着费米系统对纠缠有明显的影响。在(1,1)系统的临界温度比在(1/2,1/2)和(1/2,1)的大,根据这一点可以在相对高的温度得到纠缠,这是具有现实意义的。在(1,1)系统中,铁磁和反铁磁情况下的临界温度几乎一致,这个情况和其他两个系统是不同的。
In the 20th century, quantum mechanics expand a new area that is quantum information, and quantum entanglement is the key. Quantum entanglement is a wonderful phenomenon which presented in many subsystems of quantum systems, and because of non-local features entanglement has been used as a potentially valuable resource in quantum computing, quantum information processing and plays an important role. By the concept of Negativity, in this paper we research the property of entanglement in Heisenberg XY model and XXZ model and we get some interesting conclusions. Specific study in this article as follows:
For Heisenberg XY model with DM interaction D and magnetic field B we study the influence of different parameters to entanglement. Through calculation we known that B can change the value and the feature of entanglement at low temperature, and it affects the critical temperature based on the value of anisotropy. It is easily to find that D does not change the revival feature, and it only affects the area near critical magnetic field Bc and adjusts the value of Bc. As J increased the maximum of entanglement are all the same, no matter what the value of B and D are.
Then we study the thermal entanglement in Heisenberg XXZ model with DM interaction. Through calculation, we know that for XXZ model the anisotropy and spin can be used together to control the extent of entanglement and, in particular, to obtain large entanglement The effect of spin in both models show that it can increase the critical temperature and the negativity decreases as the spin increases. We found that the DM interaction has different effects on fermi and bose systems so it can not only excite entanglement but also affect the entanglement in different spin systems.
We investigate the ground-state and thermal entanglement in spin-1 Heisenberg XXX model and XXZ model with DM interaction, and in order to get better conclusion we study this system compared to the system of (1/2,1/2) and (1/2,1). We find that the entanglement is maximum in (1/2,1/2) and is minimum in (1/2,1) which means the fermi system has significant effect to the entanglement. We get that the critical temperature in (1,1) is biggest than that in (1/2,1/2) and (1/2,1) which has practical significance to obtain entanglement at high temperature, and in (1,1) the critical temperature of J>0 is almost the same to that of J<0 which is different form the other cases.
在磁场B存在的情况下研究了具有Dzyaloshinski-Moriya (DM)相互：作用(D)的Heisenberg XY模型,讨论了几种物理量对纠缠的影响。发现在低温时磁场B对纠缠值与纠缠特性都有一定的影响,并且还发现根据各向异性参数γ的不同从而影响临界温度的大小。通过研究发现DM相互作用不影响纠缠回升的过程,它只影响临界磁场Bc区域并且调节临界磁场Bc的大小。当J增加到一定值时,无论B和D的值多大,纠缠的最大值都是不变的。
利用Negativity的方法在Heisenberg XXZ模型中计算了具有DM相互作用的自旋间的热纠缠,分析了温度、DM相互作用、各向异性参数、耦合常数和自旋大小对纠缠的影响。通过计算两粒子之间的纠缠,发现铁磁和反铁磁情况下的纠缠情况不同：反铁磁情况下的纠缠值一般大于铁磁情况下的纠缠值。住D相对较小时,纠缠在铁磁和反铁磁情况下的值是不同的。而随着D的增加,纠缠最后会达到相同的值。在DM相互作用下,两个费米子系统的纠缠最大值大于混合系统的纠缠最大值,由此得出,DM相互作用对两个费米子系统的影响比较明显。混合系统在一个相对较小的D值就会达到纠缠最大值,然而两个费米子系统需要较强的DM相互作用才能达到纠缠最大值。
进一步研究了自旋为1的具有DM相互作用的Heisenberg XXX模型和XXZ模型的基态和热纠缠,将结果和(1/2,1/2),(1/2,1)系统的结果比较,发现纠缠值在(1/2,1/2)系统最大,在(1/2,1)系统最小,这意味着费米系统对纠缠有明显的影响。在(1,1)系统的临界温度比在(1/2,1/2)和(1/2,1)的大,根据这一点可以在相对高的温度得到纠缠,这是具有现实意义的。在(1,1)系统中,铁磁和反铁磁情况下的临界温度几乎一致,这个情况和其他两个系统是不同的。
In the 20th century, quantum mechanics expand a new area that is quantum information, and quantum entanglement is the key. Quantum entanglement is a wonderful phenomenon which presented in many subsystems of quantum systems, and because of non-local features entanglement has been used as a potentially valuable resource in quantum computing, quantum information processing and plays an important role. By the concept of Negativity, in this paper we research the property of entanglement in Heisenberg XY model and XXZ model and we get some interesting conclusions. Specific study in this article as follows:
For Heisenberg XY model with DM interaction D and magnetic field B we study the influence of different parameters to entanglement. Through calculation we known that B can change the value and the feature of entanglement at low temperature, and it affects the critical temperature based on the value of anisotropy. It is easily to find that D does not change the revival feature, and it only affects the area near critical magnetic field Bc and adjusts the value of Bc. As J increased the maximum of entanglement are all the same, no matter what the value of B and D are.
Then we study the thermal entanglement in Heisenberg XXZ model with DM interaction. Through calculation, we know that for XXZ model the anisotropy and spin can be used together to control the extent of entanglement and, in particular, to obtain large entanglement The effect of spin in both models show that it can increase the critical temperature and the negativity decreases as the spin increases. We found that the DM interaction has different effects on fermi and bose systems so it can not only excite entanglement but also affect the entanglement in different spin systems.
We investigate the ground-state and thermal entanglement in spin-1 Heisenberg XXX model and XXZ model with DM interaction, and in order to get better conclusion we study this system compared to the system of (1/2,1/2) and (1/2,1). We find that the entanglement is maximum in (1/2,1/2) and is minimum in (1/2,1) which means the fermi system has significant effect to the entanglement. We get that the critical temperature in (1,1) is biggest than that in (1/2,1/2) and (1/2,1) which has practical significance to obtain entanglement at high temperature, and in (1,1) the critical temperature of J>0 is almost the same to that of J<0 which is different form the other cases.
引文
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[2]G. J. Miburn著,郭光灿等译.费曼处理器：量子计算机简介[M].南昌：江西教育出版社.1999.
[3]张礼,葛墨林.量子力学的前沿问题[M].北京：清华大学出版社,2000.
[4]T. Jennewein, C. Simon, G. Weihs et al. Quantum cryptography with entangled photons [J]. Phy. Rev. Lett.,2000,84:4729.
[5]A. K. Ekert. Quantum cryptography based on Bell's theorem [J]. Phys. Rev. Lett.,1991, 67:661.
[6]C. H. Bennett and S. J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states [J]. Phys. Rev. Lett.,1992,69:2881.
[7]C. H. Bennett, G. Brassard, C. Crepeau et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels [J]. Phys. Rev. Lett.,1993,70: 1895.
[8]P. W. Shor. Scheme for reducing decoherence in quantum computer memory [J]. Phys. Rev. A,1995,52:R2493.
[9]J. Von Neumann. Mathematical Foundation of Quantum Mechanics [M]. Princeton:Princeton University Press,1955.
[10]C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters. Mixed-state entanglement and quantum error correction [J]. Phys. Rev. A,1996,54:3824.
[11]S. Hill and W. K. Wootters. Entanglement of a pair of quantum bits [J]. Phys. Rev. Lett., 1997,78:5022.
[12]W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits [J]. Phys. Rev. Lett.,1998,80:2245.
[13]K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein. Volume of the set of separable states [J]. Phys. Rev. A,1998,58:883.
[14]A. Peres. Separability criterion for density matrices [J]. Phys. Rev. Lett.,1996,77:1413.
[15]M. Horodecki, P. Horodecki and R. Horodecki. Separability of mixed states:necessary and sufficient conditions [J]. Phys. Lett. A,1996,223:1-8.
[16]G. Vidal and R. F. Werner. Computable measure of entanglement [J]. Phys. Rev. A,2002, 65:032314.
[17]M. Christandl and A. Winter. "Squashed entanglement"-an additive entanglement measure [J]. J. Math. Phys.,2004,45:829.
[18]Shi-Quan Su, Jun-Liang Song and Shi-Jian Gu. Local entanglement and quantum phase transition in a one-dimensional transverse field Ising model [J]. Phys. Rev. A,2006,74: 032308.
[19]S. O. Skroseth and S. D. Bartlett. Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields [J]. Phys. Rev. A,2009,80: 022316.
[20]Xiao-San Ma, An-Min Wang and Ya Cao. Entanglement evolution of three-qubit states in a quantum-critical environment [J]. Phys. Rev. B,2007,76:155327.
[21]R. Jafari, M. Kargarian, A. Langari and M. Siahatgar. Phase diagram and entanglement of the Ising model with Dzyaloshinskii-Moriya interaction [J]. Phys. Rev. B,2008,78: 214414.
[22]Ye Yeo. Teleportation via thermally entangled states of a two-qubit Heisenberg XX chain [J]. Phys. Rev. A,2002,66:062312.
[23]Min Cao and Shi-Qun Zhu. Thermal entanglement between alternate qubits of a four-qubit Heisenberg XX chain in a magnetic field [J]. Phys. Rev. A,2005,71:034311.
[24]N. Laflorencie, E. S. Soensen, Ming-Shyang Chang and I. Affleck. Boundary effects in the critical scaling of entanglement entropy in 1D systems [J]. Phys. Rev. Lett.,2006,96: 100603.
[25]Xiao-Feng Qian and Z. Song. Relation between two measures of entanglement in spin-1/2 and spinless fermion quantum chain systems [J]. Phys. Rev. A,2006,74:022302.
[26]田丽珍,杜安.自旋-3/2 Ising模型的扩展Bethe-Peierls近似计算[J].辽宁师专学报,2005,7：1.
[27]Osborne, T. J. Nielsen and A. Michael. Entanglement in a simple quantum phase transition [J]. Phys. Rev. A,2002,66:032110.
[28]A. Lakshminarayan and V. Subrahmanyam. Multipartite entanglement in a one-dimensional time-dependent Ising model [J]. Phys. Rev. A,2005,71:062334.
[29]J. Karthik, A. Sharma and A. Lakshminarayan. Entanglement, avoided crossings, and quantum chaos in an Ising model with a tilted magnetic field [J]. Phys. Rev. A,2007,75: 022304.
[30]J. Dziarmaga. Dynamics of a quantum phase transition:exact solution of the quantum Ising model [J]. Phys. Rev. Lett.,2005,95:245701.
[31]张英杰,夏云杰,满忠晓,郭光灿Ising模型的腔QED模拟和最大纠缠态的存储以及四体纠缠态的产生[J].中国科学G辑：物理学力学天文学,2009,39：494.
[32]E. Lieb, T. Schultz and D. Mattis. Two soluble models of an antiferromagnetic chain [J]. Annals of Physics,2004,16:407.
[33]Ben-Qiong Liu, Bin Shao and Jian Zou. Entanglement of two qubits coupled to an XY spin chain:role of energy current [J]. Phys. Rev. A,2009,80:062322.
[34]G. L. Kamta and A. F. Starace. Anisotropy and magnetic field effects on the entanglement of a two qubit Heisenberg XY chain [J]. Phys. Rev. Lett.,2002,88:107901.
[35]Shi-Jian Gu, Guang-Shan Tian and Hai-Qing Lin. Ground-state entanglement in the XXZ model [J]. Phys. Rev. A,2005,71:052322.
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