量子纠缠和保真度在凝聚态系统中的应用
|
摘要
量子纠缠和量子保真度是量子力学中非常重要的两个概念。在近些年迅速发展的量子信息和量子计算领域中,二者有着非常广泛的应用。而对于这两个概念的研究其实从量子理论建立初期就已经开始了。量子纠缠反映了量子世界的相干性,或然性和非定域性等最为基本的特性。保真度的概念在表征系统稳定性,研究系统演化的不可逆过程等基本问题以及对于量子混沌的刻画中都有非常重要的应用。近些年来,许多学者将这两个概念应用在凝聚态系统中得到了一些很有价值的结果。海森堡自旋链就是一个被广泛研究的凝聚态系统。本文就是利用一些有效的纠缠度来刻画一系列海森堡系统中的纠缠特性,也包括纠缠的动力学演化行为的研究,此外还将利用保真度来表征具有量子相变的系统在临界点位置的特殊性质。
文章主要内容可以分为四个部分。第一个部分就是论文中的第一章。主要介绍一些基本概念和将要用到的一些纠缠量度,其中对于负值度的介绍较为详细,包括负值度的来源、适用范围以及对其研究的一些发展情况。也较为详细的介绍了保真度概念以及保真度的敏感度。
第二部分主要包括第二章、第三章和第四章。这一部分主要是应用Concur-rence和负值度两种纠缠度来研究一些海森堡模型中的纠缠特性。其中第二章中主要研究了自旋1／2体系中的二聚型(dimerized)海森堡链以及带有外场的XY模型。对于小尺度情况给出了纠缠度的解析结果,并且利用数值计算了平均Concurrence随着系统参数的变化情况。在XY模型中主要考察了交错型外场对于纠缠的影响。以及对应不同的各向异性参数的纠缠行为。第三章主要利用负值度来研究各向同性的(1／2,1)混合自旋海森堡系统中的热纠缠问题。对于小尺度情况给出解析结果,偶数格点的情况会得到负值度和系统内能的关系式。并且分析了温度对于纠缠的影响,给出域值温度的定义和解析值。并且还考察了带有次近邻相互作用的混合自旋系统的纠缠所受到的来自于次近邻相互作用的影响。第四章中研究了自旋1系统中,具有二聚型相互作用和带有次近邻相互作用两种情况下的热纠缠。考察了不同种类相互作用以及温度对于系统纠缠的影响。
第三部分包括第五章和第六章,主要考察了纠缠的动力学演化行为。第五章中研究了两个自旋1的系统与环境耦合,利用一个典型的消相位模型作为驱动哈密顿量。并且考虑了系统初始时刻处于两种束缚(bound)纠缠态的情况,环境也选择两种不同的模型来模拟,一个是自由玻色子热库,另一个是处于热平衡的独立的自旋系统。综合使用重排和负值度两种探测量通过解析和数值的方法研究了两种束缚纠缠态在这两种环境中的动力学行为。第六章中,将环境选择为一个典型的具有量子相变的横场伊辛模型,该模型是精确可解得。考察了系统处于最大纠缠纯态和Werner混态情况下的纠缠动力学行为。重点研究了环境处于量子相变点位置时系统纠缠的演化行为,并且给出了退相干因子的解析表达式,据此得到负值度随时间演化的规律。
第四部分为论文的第七章。提出了算符保真度的概念,分析了这个保真度的特点以及处理具体问题的优势。在横场伊辛模型中得到该保真度的解析结果及其敏感度的解析结果。数值计算了敏感度随外场参量变化的行为,重点分析了敏感度及其导数在相变点位置的特殊性质。此外还考虑了带有次近邻相互作用的自旋1／2海森堡系统,选取了奇数格点情况,算符保真度的敏感度在系统基态是简并的情况仍然可以指示出临界点的位置。第七章的第二节中我们考察了(1／2,1)混合自旋XXZ海森堡系统中的基态保真度、纠缠和纯度。会发现三者在临界点都会表现出特殊行为。
Quantum entanglement and quantum fidelity are two important concepts in the quantum theory. In the field of quantum information and quantum com-putation, the two concepts play the key roles. And they are not new ones in quantum theory, some researches bout them should trace back to the beginning of the quantum theory. Quantum entanglement reflects the basic properties of the quantum world such as the quantum coherence, probability and nonlocality. The quantum fidelity was used as a measure of stability and also used to study ir-reversibility, especially it is very useful in investigating quantum chaos. Recently, the study of entanglement properties and fidelity behaviors in many-body sys-tems have attracted much attention. The Heisenberg chains are widely studied in the condensed matter field. In this thesis, we will investigate the entangle-ment properties in a series of Heisenberg chains, and also study the dynamical behavior of entanglement. On the other hand, we will use fidelity and the fidelity susceptibility to indicate quantum phase transition.
This thesis comes in four parts. The first part is Chapter 1 where we sys-tematically introduce the concepts, such as some measure of entanglement and the definition of fidelity and fidelity susceptibility. In the part, we introduce the negativity in detail.
The second part includes Chapter 2,3 and 4 where we discuss the entangle-ment properties in some Heisenberg chains by uss of concurrence and negativity. In Chapter 2, we consider the spin-1/2 dimerized system and the XY system in external magnetic field. For small sizes system, the analytical results of entangle-ment can be obtained. By numerical calculation, we study the mean concurrence versus different interaction strength. In the XY system, we consider the effects of the stagger external field on the entanglement. In Chapter 3, we study the entan-glement properties in the (1/2,1) mixed-spin chain. In this system the negativity is a very well entanglement measure. In small systems, we can give the analytical results, and in the system with even sites, we can obtain the relation between entanglement and the system energy, which is a general and important result. Thermal entanglement is considered and the threshold temperature is given. In Chapter 4, spin-1 systems are considered. In the case with next-nearest-neighbor (NNN) interaction, we numerically study the entanglement versus NNN interaci-ton, and also present how the thermal fluctuation destroy entanglement.
The third part includes Chapter 5 and 6, where we study the dynamical behaviors of entanglement. In Chapter 5, we consider two spins coupled to a environment which is characterized by a pure dephasing model. We assume the initial state of the two spins is two kinds of bound entangled states. We also choose two models to simulate the environment:one is free bosons in heat bath, and the other is a system consists of lots uncorrelated spin halves in thermal balance. Negativity and realignment are all used to characterized the dynamical properties of entanglement. In Chapter 6, we choose the Ising spin chain in a transverse field to act the environment system, which can be exactly calculated. The analytical results of the decoherence factors are given, and based on which we can obtain the expression of entanglement decaying with time. We focus on the critical point, at which the entanglement decays monotonously with time. When the two spins initially starts from a werner state, the complete disentanglement happens.
The forth part is the chapter 7. In this part we first give a new concept called operator fidelity, which is state independent. In the system of Ising spin chain in a transverse field, the susceptibility of the operator fidelity presents special behavior at the critical point, which means that the operator fidelity can be a indicator of quantum phase transition. In another example system, the spin-1/2 system with NNN interaction, we consider the odd sites cases and the susceptibility of the operator fidelity can indicate the critical point overcoming the degenerate of ground state. In the second section of this part, we consider anisotropy (1/2,1) mixed-spin chain which also has quantum criticality. The fidelity, entanglement and purity are studied in this system. The three quantum all present special properties at the critical point.
文章主要内容可以分为四个部分。第一个部分就是论文中的第一章。主要介绍一些基本概念和将要用到的一些纠缠量度,其中对于负值度的介绍较为详细,包括负值度的来源、适用范围以及对其研究的一些发展情况。也较为详细的介绍了保真度概念以及保真度的敏感度。
第二部分主要包括第二章、第三章和第四章。这一部分主要是应用Concur-rence和负值度两种纠缠度来研究一些海森堡模型中的纠缠特性。其中第二章中主要研究了自旋1／2体系中的二聚型(dimerized)海森堡链以及带有外场的XY模型。对于小尺度情况给出了纠缠度的解析结果,并且利用数值计算了平均Concurrence随着系统参数的变化情况。在XY模型中主要考察了交错型外场对于纠缠的影响。以及对应不同的各向异性参数的纠缠行为。第三章主要利用负值度来研究各向同性的(1／2,1)混合自旋海森堡系统中的热纠缠问题。对于小尺度情况给出解析结果,偶数格点的情况会得到负值度和系统内能的关系式。并且分析了温度对于纠缠的影响,给出域值温度的定义和解析值。并且还考察了带有次近邻相互作用的混合自旋系统的纠缠所受到的来自于次近邻相互作用的影响。第四章中研究了自旋1系统中,具有二聚型相互作用和带有次近邻相互作用两种情况下的热纠缠。考察了不同种类相互作用以及温度对于系统纠缠的影响。
第三部分包括第五章和第六章,主要考察了纠缠的动力学演化行为。第五章中研究了两个自旋1的系统与环境耦合,利用一个典型的消相位模型作为驱动哈密顿量。并且考虑了系统初始时刻处于两种束缚(bound)纠缠态的情况,环境也选择两种不同的模型来模拟,一个是自由玻色子热库,另一个是处于热平衡的独立的自旋系统。综合使用重排和负值度两种探测量通过解析和数值的方法研究了两种束缚纠缠态在这两种环境中的动力学行为。第六章中,将环境选择为一个典型的具有量子相变的横场伊辛模型,该模型是精确可解得。考察了系统处于最大纠缠纯态和Werner混态情况下的纠缠动力学行为。重点研究了环境处于量子相变点位置时系统纠缠的演化行为,并且给出了退相干因子的解析表达式,据此得到负值度随时间演化的规律。
第四部分为论文的第七章。提出了算符保真度的概念,分析了这个保真度的特点以及处理具体问题的优势。在横场伊辛模型中得到该保真度的解析结果及其敏感度的解析结果。数值计算了敏感度随外场参量变化的行为,重点分析了敏感度及其导数在相变点位置的特殊性质。此外还考虑了带有次近邻相互作用的自旋1／2海森堡系统,选取了奇数格点情况,算符保真度的敏感度在系统基态是简并的情况仍然可以指示出临界点的位置。第七章的第二节中我们考察了(1／2,1)混合自旋XXZ海森堡系统中的基态保真度、纠缠和纯度。会发现三者在临界点都会表现出特殊行为。
Quantum entanglement and quantum fidelity are two important concepts in the quantum theory. In the field of quantum information and quantum com-putation, the two concepts play the key roles. And they are not new ones in quantum theory, some researches bout them should trace back to the beginning of the quantum theory. Quantum entanglement reflects the basic properties of the quantum world such as the quantum coherence, probability and nonlocality. The quantum fidelity was used as a measure of stability and also used to study ir-reversibility, especially it is very useful in investigating quantum chaos. Recently, the study of entanglement properties and fidelity behaviors in many-body sys-tems have attracted much attention. The Heisenberg chains are widely studied in the condensed matter field. In this thesis, we will investigate the entangle-ment properties in a series of Heisenberg chains, and also study the dynamical behavior of entanglement. On the other hand, we will use fidelity and the fidelity susceptibility to indicate quantum phase transition.
This thesis comes in four parts. The first part is Chapter 1 where we sys-tematically introduce the concepts, such as some measure of entanglement and the definition of fidelity and fidelity susceptibility. In the part, we introduce the negativity in detail.
The second part includes Chapter 2,3 and 4 where we discuss the entangle-ment properties in some Heisenberg chains by uss of concurrence and negativity. In Chapter 2, we consider the spin-1/2 dimerized system and the XY system in external magnetic field. For small sizes system, the analytical results of entangle-ment can be obtained. By numerical calculation, we study the mean concurrence versus different interaction strength. In the XY system, we consider the effects of the stagger external field on the entanglement. In Chapter 3, we study the entan-glement properties in the (1/2,1) mixed-spin chain. In this system the negativity is a very well entanglement measure. In small systems, we can give the analytical results, and in the system with even sites, we can obtain the relation between entanglement and the system energy, which is a general and important result. Thermal entanglement is considered and the threshold temperature is given. In Chapter 4, spin-1 systems are considered. In the case with next-nearest-neighbor (NNN) interaction, we numerically study the entanglement versus NNN interaci-ton, and also present how the thermal fluctuation destroy entanglement.
The third part includes Chapter 5 and 6, where we study the dynamical behaviors of entanglement. In Chapter 5, we consider two spins coupled to a environment which is characterized by a pure dephasing model. We assume the initial state of the two spins is two kinds of bound entangled states. We also choose two models to simulate the environment:one is free bosons in heat bath, and the other is a system consists of lots uncorrelated spin halves in thermal balance. Negativity and realignment are all used to characterized the dynamical properties of entanglement. In Chapter 6, we choose the Ising spin chain in a transverse field to act the environment system, which can be exactly calculated. The analytical results of the decoherence factors are given, and based on which we can obtain the expression of entanglement decaying with time. We focus on the critical point, at which the entanglement decays monotonously with time. When the two spins initially starts from a werner state, the complete disentanglement happens.
The forth part is the chapter 7. In this part we first give a new concept called operator fidelity, which is state independent. In the system of Ising spin chain in a transverse field, the susceptibility of the operator fidelity presents special behavior at the critical point, which means that the operator fidelity can be a indicator of quantum phase transition. In another example system, the spin-1/2 system with NNN interaction, we consider the odd sites cases and the susceptibility of the operator fidelity can indicate the critical point overcoming the degenerate of ground state. In the second section of this part, we consider anisotropy (1/2,1) mixed-spin chain which also has quantum criticality. The fidelity, entanglement and purity are studied in this system. The three quantum all present special properties at the critical point.
引文
[1]M. A. Turing, On computable numbers, with an application to the Entschei-dungsproblem, Proc. Lond. Math. Soc. (Ser.2) 42,230-265 (1936).
[2]Paul Benioff, Phys. Rev. Lett.48,1581 (1982).
[3]RFeynman, International Journal of Theoretical Physics 21,467 (1982).
[4]S. L. Braunstein et al., Phys. Rev. Lett.83,1054 (1999); N. Linden and S. Popescu, Phys. Rev. Lett.87,047901 (2001); G. Vidal, Phys. Rev. Lett.91, 147902 (2003); Chao-Yang Lu, Daniel E. Browne, Tao Yang, and Jian-Wei Pan, Phys. Rev. Lett.99,250504 (2007).
[5]M. Mohseni et al., Phys. Rev. Lett.91,187903 (2003); M. S. Tame et al., Phys. Rev. Lett.98,140501 (2007); P. G. Kwiat et al., J. Mod. Opt.47,257 (2000); P. Walther et al., Nature (London) 434,169 (2005); R. Prevedel et al., Nature (London) 445,65 (2007).
[6]E. Schrodinger, Naturwiss.48,807 (1935).
[7]D Bohm,《量子理论》(侯德彭译,北京商务印书馆,1982).
[8]张永德,《量子信息物理原理》(科学出版社2005年)
[9]J. S. Bell, Physics,1195, (1964).
[10]A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London) 416,608 (2002).
[11]T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66,032110 (2002).
[12]S. J. Gu, H. Q. Lin, and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[13]Y. Chen, P. Zanardi, Z. D. Wang, and F. C. Zhang, New J. of Phys.897 (2006).
[14]Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters, Phys. Rev. A 54,3824 (1996).
[15]S. Hill and W. K. Wootters, Phys. Rev. Lett.78,5022 (1997).
[16]W. K. Wootters, Phys. Rev. Lett.80,2245 (1997).
[17]Kevin M.0'Connor and William K. Wootters, Phys. Rev. A 63,052302 (2001).
[18]C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett.70,1895 (1993).
[19]C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, W. K. Wootters, Phys. Rev. Lett.76,722 (1996).
[20]C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53,2046 (1996).
[21]Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, Phys. Rev. Lett.80,5239 (1998).
[22]B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein, Phys. Rev. A 61,062302 (2000).
[23]David P. DiVincenzo, Peter W. Shor, John A. Smolin, Barbara M. Terhal, and Ashish V. Thapliyal, Phys. Rev. A 61,062312 (2000).
[24]K. Chen and L. A. Wu, Quantum Inf. Comput.3,193 (2003).
[25]Oliver Rudolph, Quantum Information Processing,4219 (2005).
[26]Thomas Gorina, Tomaz Prosen, Thomas H. Seligman, Marko Znidaric, Physics Reports 435,33-156 (2006)
[27]Asher Peres, Phys. Rev. A 30,1610 (1984).
[28]H.M. Pastawski et al, Phys. Rev. Lett.75,4310 (1995); G. Usaj et al, Mol. Phys.95,1229 (1998); P.R. Levstein et al., J. Chem. Phys.108,2718 (1998); H.M. Pastawski et al., Physica A 283,166 (2000).
[29]R. Jalabert and H. Pastawski, Phys. Rev. Lett.86,2490(2001).
[30]W. Rhim, A. Pines, and J.Waugh, Phys. Rev. Lett.25,218 (1970).
[31]S. Zhang, B. Meier, and R. Ernst, Phys. Rev. Lett.69,2149 (1992).
[32]T. Prosen, Phys. Rev. E 65,045206 (2002); T. Prosen, M. Znidaric, J. Phys. A 35,1455 (2002).
[33]Z. Karkuszewski, C. Jarzynski, and W. Zurek, Phys. Rev. Lett.89,170405 (2002); T. Gorin, T. Prosen, T.H. Seligman, and W.T. Strunz, Phys. Rev. A 70,042105 (2004).
[34]H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96,140604 (2006).
[35]P. Zanardi and N. Paunkovic, Phys. Rev. E 74,031123 (2006).
[36]H. Q. Zhou and J. P. Barjaktarevic, cond-mat/0701608.
[37]W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. E 76,022101 (2007).
[38]P. Zanardi and N. Paunkovic Phys. Rev. E 74,031123 (2006); Cozzini, P. Giorda, and P. Zanardi, Phys. Rev. B 75,014439 (2007); M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B 76,104420 (2007); P. Buonsante and A. Vezzani, Phys. Rev. Lett.98,110601 (2007); Min-Fong Yang, Phys. Rev. B 74,180403 (2007).
[39]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum In-formation (Cambridge University Press, Cambridge,2000).
[40]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[41]J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett.23, 880 (1969).
[42]E. Schrodinger, Naturwiss.48,807 (1935).
[43]R. Horodecki and P. Horodecki, Phys. Lett. A 194,147 (1994); R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 210,377 (1996); R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Rev. A 54,1838 (1996).
[44]Asher Peres, Phys. Rev. Lett.77,1413 (1996).
[45]Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, Phys. Lett. A 223 (1996) 1-8.
[46]G. Alber, et al Quantum Information, (Springer Tracts in Modern Physics Vol.173).
[47]E. Strφmer, Acta. Math.110,233 (1963); S. L. Woronowicz, Rep. Math. Phys.10165 (1976).
[48]G. Vidal and R. F. Werner, Phys. Rev. A 65,032314 (2002).
[49]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum In-formation (Cambridge University Press, Cambridge,2000).
[50]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[51]Zhe Sun and Xiaoguang Wang, Phys. Rev. A 75,062312 (2007).
[52]E. M. Rains, Phys. Rev. A 60,179 (1999).
[53]M. Khasin, R. Kosloff, and D. Steinitz, Phys. Rev. A 75,052325 (2007).
[54]J. Schliemann, Phys. Rev. A 68,012309 (2003).
[55]J. Schliemann, Phys. Rev. A 72,012307 (2005).
[56]H.P. Breuer, J. Phys. A 38,9019 (2005).
[57]H.P. Breuer, Phys. Rev. A 71,062330 (2005).
[58]Huaixiang Huang, Xiaoguang Wang, Zhe Sun, and Guohong Yang, Phyica A 3872736-2744 (2008).
[59]M. T. Batchelor and M. N. Barber, J. Phys. A 2315 (1990), A. Klumper, J. Phys. A 23,809 (1990).
[60]G. M. Zhang, and X. Wang, J. Phys. A:Math 39,8515, (2006).
[61]M. A. Nielsen, Ph. D thesis, University of Mexico,1998, quant-ph/0011036;
[62]M. C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett.87,017901 (2001).
[63]X. Wang, Phys. Rev. A 64,012313 (2001); Phys. Lett. A 281,101 (2001).
[64]X. Wang, H. Fu, and A. I. Solomon, J. Phys. A:Math. Gen.34,11307(2001); X. Wang and K. Mφlmer, Eur. Phys. J. D 18,385(2002).
[65]X. Wang and P. Zanardi, Phys. Lett. A 301,1 (2002); X. Wang, Phys. Rev. A 66,044305 (2002).
[66]S. Bose and V. Vedral, Phys. Rev. A 61,040101 (2000).
[67]K. M. O'Connor and W. K. Wootters, Phys. Rev. A 63,0520302 (2001).
[68]Y. Sun, Y. G. Chen, and H. Chen, Phys. Rev. A 68,044301 (2003).
[69]U. Glaser, H. Biittner, and H. Fehske, Phys. Rev. A 68,032318 (2003).
[70]D. V. Khveshchenko, Phys. Rev. B 68,193307 (2003).
[71]L. Zhou, H. S. Song, Y. Q. Guo, and C. Li, Phys. Rev. A 68,024301 (2003).
[72]F. Verstraete, M. Popp, and J. I. Cirac, Phys. Rev. Lett.92,027901 (2004).
[73]J. Vidal, G. Palacios, and R. Mosseri, Phys. Rev. A 69,022107 (2004).
[74]N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett.92,073602 (2004).
[75]S. Ghose, T. F. Rosenbaum, G. Aeppli, and S. N. Coppersmith, Nature (London) 425,48 (2003).
[76]J. Schliemann, Phys. Rev. A 68,012309 (2003).
[77]G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev Phys. Rev. Lett.90,227902 (2003).
[78]H. A. Bethe, Z. Phys.71,205 (1931).
[79]C. N. Yang and C. P. Yang, Phys. Rev.150,321 (1966).
[80]S. J. Gu, H. Q. Lin, and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[81]X. Wang, Phys. Lett. A 329,439 (2004).
[82]D. Kouzoudis, J. Magn. Magn. Mater.173,259 (1997); ibid 189,366 (1998).
[83]K. Barwinkel, H.-J. Schmidt, and J. Schnack, J. Magn. Magn. Mater.220, 227 (2000).
[84]H. Q. Lin, Phys. Rev. B 42,6561 (1990).
[85]X. Wang, Phys. Rev. E 69,066118 (2004).
[86]S. Bose, Phys. Rev. Letts.91,207901 (2003); V. Subrahmanyam,Phys. Rev. A 69,034304 (2004); M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Phys. Rev. Letts.92,187902 (2004); Y. Li, T. Shi, Z. Song, and C. P. Sun, Phys. Rev. A 71,022301 (2005).
[87]F. C. Alcarazy and A. L. Malvezzi, J. Phys. A:Math. Gen.30,767-778 (1997).
[88]S. K. Pati, S. Ramasesha, and D. Sen, Phys. Rev. B 55,8894 (1997).
[89]S. Brehmer, H. J. Mikeska, and S. Yamanoto, J. Phys.:Condens. Matter 9, 3921 (1997).
[90]S. Yamamoto, T. Fukui, K. Maisinger, and U. Schollwock, J. Phys.:Condens. Matter 10,11033 (1998).
[91]S. Yamamoto and T. Fukui, Phys. Rev. B 57, R14008 (1998); Noboru Fukushima, Andreas Honecker, Stefan Wessel, and Wolfram Brenig, Phys. Rev. B 69,174430 (2004).
[92]T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66,032110 (2002).
[93]A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature 416,608 (2002).
[94]M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett.70,3651 (1993).
[95]J. W. Bray et al., in Extended Linear Chain Compounds, edited by J. S. Miller (Plenum, New Youk,1993), Vol.3, pp.353-415.
[96]I. Bose and E. Chattopadhyay, Phys. Rev. A 66,062320 (2002).
[97]Yasuo Narumi, Masayuki Hagiwara, Masanori Kohno, and Koichi Kindo, Phys. Rev. Lett.86,324 (2001).
[98]Weihong Zheng and J. Oitmaa, Phys. Rev. B 67,224421 (2003).
[99]K. Maisinger, U. Schollwock, S. Brehmer, H. J. Mikeska, and Shoji Ya-mamoto, Phys. Rev. B 58,5908 (1998).
[100]Xiao-Feng Qian, Tao Shi, Ying Li, Z. Song, and C. P. Sun, Phys. Rev. A 72,012333 (2005).
[101]Indrani Bose and Emily Chattopadhyay, Phys. Rev. A 66,062320 (2002).
[102]F. D. M. Haldane, Phys. Lett. A 93,464 (1983); F. D. M. Haldane, Phys. Rev. Lett.50,1153 (1983).
[103]I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett.59,799 (1987).
[104]P. Millet, F. Mila, F. C. Zhang, M. Mambrini, A. B. Van Oosten, V. A. Pashchenko, A. Sulpice, and A. Stepanov, Phys. Rev. Lett.83,4176 (1999).
[105]J. Z. Lou, T. Xiang, and Z. B. Su, Phys. Rev. Lett.85,2380 (2000).
[106]Schollwock, T. Jolicoeur, and T. Garel, Phys. Rev. B 53,3304 (1996).
[107]H. Fan, V. Korepin, and V. Roychowdhury, Phys. Rev. Lett.93,227203 (2004).
[108]F. Verstraete, M. A. Martin-Delgado, and J. I. Cirac, Phys. Rev. Lett.92, 087201 (2004).
[109]L. Zhou, X. X. Yi, H. S. Song, and Y. Q. Quo, Journal of Optics B 6 378-382 (2004).
[110]X. Wang, H. B. Li, Z. Sun, and Y. Q. Li, J. Phys. A:Math. Gen.38,8703 (2005).
[111]A. Kolezhuk, R. Roth, and U. Schollwock, Phys. Rev. Lett.77,5142 (1996).
[112]T. Yu and J. H. Eberly, Phys. Rev. Lett.93,140404 (2004).
[113]M. S. Zubairy, G. S. Agarwal, and M. O. Scully, Phys. Rev. A 70,012316 (2004).
[114]K. Roszak and P. Machnikowski Phys. Rev. A 73,022313 (2006).
[115]P. Horodecki, M. Horodecki and R. Horodecki, Phys. Rev. Lett.82, 1056(1999).
[116]C. P. Sun, Phys. Rev. A 48,898 (1993); C. P. Sun, H. Zhan and X. F. Liu, Phys. Rev. A 58,1810 (1998).
[117]Y.B. Gao and C.P. Sun, Phys. Rev. E 75,011105 (2007).
[118]J. Wei and E. Norman, J. Math. Phys. A 4,575 (1963).
[119]C. P. Sun, X. X. Yi, S. R. Zhao, L. Zhang and C. Wang, Quantum Semiclass. Opt 9 (1997) 119-129.
[120]Klaus Hornberger, quant-ph/0612118.
[121]C. P. Sun, X. F. Liu, D. L. Zhou and S. X. Yu, Eur. Phys. J. D 17,8592 (2001).
[122]V.Coffman, J.Kundu and W.K.Wootters, Phys. Rev. A 61,052306 (2000).
[123]C. H. Bennett et al., Phys. Rev. Lett.82,5385 (1999).
[124]W. H. Zurek, Phys. Rev. D 26,1862 (1982).
[125]F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Phys. Rev. A 72,052113 (2005).
[126]H. T. Quan, Z.Song, P. Zanardi, and C. P. Sun, Phys. Rev. Lett.96,140604 (2006).
[127]S. Sachdev, Quantum Phase Transition (Cambridge University Press, Cam-bridge England,1999)
[128]Clive Emary and Tobias Brandes, Phys. Rev. Lett.90,044101 (2003).
[129]Joseph Emerson, Yaakov S.Weinstein, Seth Lloyd and D.G. Cory, Phys. Rev. Lett.89,284102 (2002).
[130]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[131]L. Derkacz and L. Jakobczyk, Phys. Rev. A 74,032313(2006).
[132]Fernando Martin Cucchietti, Sonia Fernandez Vidal and Juan Pablo Paz, Phys. Rev. A 75,032337 (2007).
[133]C.A. Sackett et al., Nature (London),404,256 (2000).
[134]H. Q. Zhou, R. Orus, G. Vidal, cond-mat/0709.4596.
[135]T. Prosen, T. H. Seligman, and M. Znidaric, Prog. Theor. Phys. Suppl. 150,200 (2003).
[136]P. Zanardi, D.A. Lidar, Phys. Rev. A 70 (2004) 012315.
[137]C. Dankert, Efficient simulation of random quantum states and operators, Math. Thesis, University of Waterloo, quant-ph/0512217.
[138]A. Ambainis and J. Emerson, quant-ph/0701126.
[139]L. H. Pedersen, N. M. Mφller, K. Mφlmer, Phys. Lett. A 367,47 (2007).
[140]C. K. Majumdar and D. K. Ghosh, J. Math. Phys.10,1388 (1969); J. Phys. C 3,911 (1970).
[141]S. Chen, L. Wang, S. J. Gu, and Y. P. Wang, Phys. Rev. E 76,061108 (2007).
[142]A. Uhlmann, Rep. Math. Phys.9,273 (1976); R. Jozsa, J. Mod. Opt.41, 2315 (1994).
[143]W. Dur, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett, 87,137901 (2001); M. S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A,67,012306 (2003).
[144]P. Zanardi, and C. Z. L. Faoro, Phys. Rev. A 62030301 (2000); P. Zanardi, Phys. Rev. A 63040304 (2001);X. Wang, and B. C. Sanders, D. W. Berry, Phys. Rev. A 67042323 (2003).
[145]Z. H. Ma and X. Wang, Phys. Rev. A 75,014304 (2007).
[146]F.C. Alcaraz and A.L. Malvezzi, J. Phys. A 30,767 (1997). T. Ono, T. Nishimura, M. Katsumua, T. Morita, and M. Sugimoto, J. Phys. Soc. Jpn. 66,2576 (1997).
[147]Guang-Shan Tian and Hai-Qing Lin, Phys. Rev. B 66,224408 (2002).
[148]Zhe Sun, Xiaoguang Wang. AnZi Hu and You-Quan Li, Physica A 370 (2006) 483-500.
[149]Shi-Jian Gu, Hai-Qing Lin, and You-Quan Li, Phys. Rev. A 68,042330 (2003).
[150]Xiaoguang Wang and Shi-Jian Gu, J. Phys. A:Math. Theor.40 (2007) 10759-10767.
[2]Paul Benioff, Phys. Rev. Lett.48,1581 (1982).
[3]RFeynman, International Journal of Theoretical Physics 21,467 (1982).
[4]S. L. Braunstein et al., Phys. Rev. Lett.83,1054 (1999); N. Linden and S. Popescu, Phys. Rev. Lett.87,047901 (2001); G. Vidal, Phys. Rev. Lett.91, 147902 (2003); Chao-Yang Lu, Daniel E. Browne, Tao Yang, and Jian-Wei Pan, Phys. Rev. Lett.99,250504 (2007).
[5]M. Mohseni et al., Phys. Rev. Lett.91,187903 (2003); M. S. Tame et al., Phys. Rev. Lett.98,140501 (2007); P. G. Kwiat et al., J. Mod. Opt.47,257 (2000); P. Walther et al., Nature (London) 434,169 (2005); R. Prevedel et al., Nature (London) 445,65 (2007).
[6]E. Schrodinger, Naturwiss.48,807 (1935).
[7]D Bohm,《量子理论》(侯德彭译,北京商务印书馆,1982).
[8]张永德,《量子信息物理原理》(科学出版社2005年)
[9]J. S. Bell, Physics,1195, (1964).
[10]A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London) 416,608 (2002).
[11]T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66,032110 (2002).
[12]S. J. Gu, H. Q. Lin, and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[13]Y. Chen, P. Zanardi, Z. D. Wang, and F. C. Zhang, New J. of Phys.897 (2006).
[14]Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters, Phys. Rev. A 54,3824 (1996).
[15]S. Hill and W. K. Wootters, Phys. Rev. Lett.78,5022 (1997).
[16]W. K. Wootters, Phys. Rev. Lett.80,2245 (1997).
[17]Kevin M.0'Connor and William K. Wootters, Phys. Rev. A 63,052302 (2001).
[18]C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett.70,1895 (1993).
[19]C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, W. K. Wootters, Phys. Rev. Lett.76,722 (1996).
[20]C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53,2046 (1996).
[21]Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, Phys. Rev. Lett.80,5239 (1998).
[22]B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein, Phys. Rev. A 61,062302 (2000).
[23]David P. DiVincenzo, Peter W. Shor, John A. Smolin, Barbara M. Terhal, and Ashish V. Thapliyal, Phys. Rev. A 61,062312 (2000).
[24]K. Chen and L. A. Wu, Quantum Inf. Comput.3,193 (2003).
[25]Oliver Rudolph, Quantum Information Processing,4219 (2005).
[26]Thomas Gorina, Tomaz Prosen, Thomas H. Seligman, Marko Znidaric, Physics Reports 435,33-156 (2006)
[27]Asher Peres, Phys. Rev. A 30,1610 (1984).
[28]H.M. Pastawski et al, Phys. Rev. Lett.75,4310 (1995); G. Usaj et al, Mol. Phys.95,1229 (1998); P.R. Levstein et al., J. Chem. Phys.108,2718 (1998); H.M. Pastawski et al., Physica A 283,166 (2000).
[29]R. Jalabert and H. Pastawski, Phys. Rev. Lett.86,2490(2001).
[30]W. Rhim, A. Pines, and J.Waugh, Phys. Rev. Lett.25,218 (1970).
[31]S. Zhang, B. Meier, and R. Ernst, Phys. Rev. Lett.69,2149 (1992).
[32]T. Prosen, Phys. Rev. E 65,045206 (2002); T. Prosen, M. Znidaric, J. Phys. A 35,1455 (2002).
[33]Z. Karkuszewski, C. Jarzynski, and W. Zurek, Phys. Rev. Lett.89,170405 (2002); T. Gorin, T. Prosen, T.H. Seligman, and W.T. Strunz, Phys. Rev. A 70,042105 (2004).
[34]H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96,140604 (2006).
[35]P. Zanardi and N. Paunkovic, Phys. Rev. E 74,031123 (2006).
[36]H. Q. Zhou and J. P. Barjaktarevic, cond-mat/0701608.
[37]W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. E 76,022101 (2007).
[38]P. Zanardi and N. Paunkovic Phys. Rev. E 74,031123 (2006); Cozzini, P. Giorda, and P. Zanardi, Phys. Rev. B 75,014439 (2007); M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B 76,104420 (2007); P. Buonsante and A. Vezzani, Phys. Rev. Lett.98,110601 (2007); Min-Fong Yang, Phys. Rev. B 74,180403 (2007).
[39]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum In-formation (Cambridge University Press, Cambridge,2000).
[40]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[41]J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett.23, 880 (1969).
[42]E. Schrodinger, Naturwiss.48,807 (1935).
[43]R. Horodecki and P. Horodecki, Phys. Lett. A 194,147 (1994); R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 210,377 (1996); R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Rev. A 54,1838 (1996).
[44]Asher Peres, Phys. Rev. Lett.77,1413 (1996).
[45]Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, Phys. Lett. A 223 (1996) 1-8.
[46]G. Alber, et al Quantum Information, (Springer Tracts in Modern Physics Vol.173).
[47]E. Strφmer, Acta. Math.110,233 (1963); S. L. Woronowicz, Rep. Math. Phys.10165 (1976).
[48]G. Vidal and R. F. Werner, Phys. Rev. A 65,032314 (2002).
[49]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum In-formation (Cambridge University Press, Cambridge,2000).
[50]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[51]Zhe Sun and Xiaoguang Wang, Phys. Rev. A 75,062312 (2007).
[52]E. M. Rains, Phys. Rev. A 60,179 (1999).
[53]M. Khasin, R. Kosloff, and D. Steinitz, Phys. Rev. A 75,052325 (2007).
[54]J. Schliemann, Phys. Rev. A 68,012309 (2003).
[55]J. Schliemann, Phys. Rev. A 72,012307 (2005).
[56]H.P. Breuer, J. Phys. A 38,9019 (2005).
[57]H.P. Breuer, Phys. Rev. A 71,062330 (2005).
[58]Huaixiang Huang, Xiaoguang Wang, Zhe Sun, and Guohong Yang, Phyica A 3872736-2744 (2008).
[59]M. T. Batchelor and M. N. Barber, J. Phys. A 2315 (1990), A. Klumper, J. Phys. A 23,809 (1990).
[60]G. M. Zhang, and X. Wang, J. Phys. A:Math 39,8515, (2006).
[61]M. A. Nielsen, Ph. D thesis, University of Mexico,1998, quant-ph/0011036;
[62]M. C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett.87,017901 (2001).
[63]X. Wang, Phys. Rev. A 64,012313 (2001); Phys. Lett. A 281,101 (2001).
[64]X. Wang, H. Fu, and A. I. Solomon, J. Phys. A:Math. Gen.34,11307(2001); X. Wang and K. Mφlmer, Eur. Phys. J. D 18,385(2002).
[65]X. Wang and P. Zanardi, Phys. Lett. A 301,1 (2002); X. Wang, Phys. Rev. A 66,044305 (2002).
[66]S. Bose and V. Vedral, Phys. Rev. A 61,040101 (2000).
[67]K. M. O'Connor and W. K. Wootters, Phys. Rev. A 63,0520302 (2001).
[68]Y. Sun, Y. G. Chen, and H. Chen, Phys. Rev. A 68,044301 (2003).
[69]U. Glaser, H. Biittner, and H. Fehske, Phys. Rev. A 68,032318 (2003).
[70]D. V. Khveshchenko, Phys. Rev. B 68,193307 (2003).
[71]L. Zhou, H. S. Song, Y. Q. Guo, and C. Li, Phys. Rev. A 68,024301 (2003).
[72]F. Verstraete, M. Popp, and J. I. Cirac, Phys. Rev. Lett.92,027901 (2004).
[73]J. Vidal, G. Palacios, and R. Mosseri, Phys. Rev. A 69,022107 (2004).
[74]N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett.92,073602 (2004).
[75]S. Ghose, T. F. Rosenbaum, G. Aeppli, and S. N. Coppersmith, Nature (London) 425,48 (2003).
[76]J. Schliemann, Phys. Rev. A 68,012309 (2003).
[77]G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev Phys. Rev. Lett.90,227902 (2003).
[78]H. A. Bethe, Z. Phys.71,205 (1931).
[79]C. N. Yang and C. P. Yang, Phys. Rev.150,321 (1966).
[80]S. J. Gu, H. Q. Lin, and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[81]X. Wang, Phys. Lett. A 329,439 (2004).
[82]D. Kouzoudis, J. Magn. Magn. Mater.173,259 (1997); ibid 189,366 (1998).
[83]K. Barwinkel, H.-J. Schmidt, and J. Schnack, J. Magn. Magn. Mater.220, 227 (2000).
[84]H. Q. Lin, Phys. Rev. B 42,6561 (1990).
[85]X. Wang, Phys. Rev. E 69,066118 (2004).
[86]S. Bose, Phys. Rev. Letts.91,207901 (2003); V. Subrahmanyam,Phys. Rev. A 69,034304 (2004); M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Phys. Rev. Letts.92,187902 (2004); Y. Li, T. Shi, Z. Song, and C. P. Sun, Phys. Rev. A 71,022301 (2005).
[87]F. C. Alcarazy and A. L. Malvezzi, J. Phys. A:Math. Gen.30,767-778 (1997).
[88]S. K. Pati, S. Ramasesha, and D. Sen, Phys. Rev. B 55,8894 (1997).
[89]S. Brehmer, H. J. Mikeska, and S. Yamanoto, J. Phys.:Condens. Matter 9, 3921 (1997).
[90]S. Yamamoto, T. Fukui, K. Maisinger, and U. Schollwock, J. Phys.:Condens. Matter 10,11033 (1998).
[91]S. Yamamoto and T. Fukui, Phys. Rev. B 57, R14008 (1998); Noboru Fukushima, Andreas Honecker, Stefan Wessel, and Wolfram Brenig, Phys. Rev. B 69,174430 (2004).
[92]T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66,032110 (2002).
[93]A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature 416,608 (2002).
[94]M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett.70,3651 (1993).
[95]J. W. Bray et al., in Extended Linear Chain Compounds, edited by J. S. Miller (Plenum, New Youk,1993), Vol.3, pp.353-415.
[96]I. Bose and E. Chattopadhyay, Phys. Rev. A 66,062320 (2002).
[97]Yasuo Narumi, Masayuki Hagiwara, Masanori Kohno, and Koichi Kindo, Phys. Rev. Lett.86,324 (2001).
[98]Weihong Zheng and J. Oitmaa, Phys. Rev. B 67,224421 (2003).
[99]K. Maisinger, U. Schollwock, S. Brehmer, H. J. Mikeska, and Shoji Ya-mamoto, Phys. Rev. B 58,5908 (1998).
[100]Xiao-Feng Qian, Tao Shi, Ying Li, Z. Song, and C. P. Sun, Phys. Rev. A 72,012333 (2005).
[101]Indrani Bose and Emily Chattopadhyay, Phys. Rev. A 66,062320 (2002).
[102]F. D. M. Haldane, Phys. Lett. A 93,464 (1983); F. D. M. Haldane, Phys. Rev. Lett.50,1153 (1983).
[103]I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett.59,799 (1987).
[104]P. Millet, F. Mila, F. C. Zhang, M. Mambrini, A. B. Van Oosten, V. A. Pashchenko, A. Sulpice, and A. Stepanov, Phys. Rev. Lett.83,4176 (1999).
[105]J. Z. Lou, T. Xiang, and Z. B. Su, Phys. Rev. Lett.85,2380 (2000).
[106]Schollwock, T. Jolicoeur, and T. Garel, Phys. Rev. B 53,3304 (1996).
[107]H. Fan, V. Korepin, and V. Roychowdhury, Phys. Rev. Lett.93,227203 (2004).
[108]F. Verstraete, M. A. Martin-Delgado, and J. I. Cirac, Phys. Rev. Lett.92, 087201 (2004).
[109]L. Zhou, X. X. Yi, H. S. Song, and Y. Q. Quo, Journal of Optics B 6 378-382 (2004).
[110]X. Wang, H. B. Li, Z. Sun, and Y. Q. Li, J. Phys. A:Math. Gen.38,8703 (2005).
[111]A. Kolezhuk, R. Roth, and U. Schollwock, Phys. Rev. Lett.77,5142 (1996).
[112]T. Yu and J. H. Eberly, Phys. Rev. Lett.93,140404 (2004).
[113]M. S. Zubairy, G. S. Agarwal, and M. O. Scully, Phys. Rev. A 70,012316 (2004).
[114]K. Roszak and P. Machnikowski Phys. Rev. A 73,022313 (2006).
[115]P. Horodecki, M. Horodecki and R. Horodecki, Phys. Rev. Lett.82, 1056(1999).
[116]C. P. Sun, Phys. Rev. A 48,898 (1993); C. P. Sun, H. Zhan and X. F. Liu, Phys. Rev. A 58,1810 (1998).
[117]Y.B. Gao and C.P. Sun, Phys. Rev. E 75,011105 (2007).
[118]J. Wei and E. Norman, J. Math. Phys. A 4,575 (1963).
[119]C. P. Sun, X. X. Yi, S. R. Zhao, L. Zhang and C. Wang, Quantum Semiclass. Opt 9 (1997) 119-129.
[120]Klaus Hornberger, quant-ph/0612118.
[121]C. P. Sun, X. F. Liu, D. L. Zhou and S. X. Yu, Eur. Phys. J. D 17,8592 (2001).
[122]V.Coffman, J.Kundu and W.K.Wootters, Phys. Rev. A 61,052306 (2000).
[123]C. H. Bennett et al., Phys. Rev. Lett.82,5385 (1999).
[124]W. H. Zurek, Phys. Rev. D 26,1862 (1982).
[125]F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Phys. Rev. A 72,052113 (2005).
[126]H. T. Quan, Z.Song, P. Zanardi, and C. P. Sun, Phys. Rev. Lett.96,140604 (2006).
[127]S. Sachdev, Quantum Phase Transition (Cambridge University Press, Cam-bridge England,1999)
[128]Clive Emary and Tobias Brandes, Phys. Rev. Lett.90,044101 (2003).
[129]Joseph Emerson, Yaakov S.Weinstein, Seth Lloyd and D.G. Cory, Phys. Rev. Lett.89,284102 (2002).
[130]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[131]L. Derkacz and L. Jakobczyk, Phys. Rev. A 74,032313(2006).
[132]Fernando Martin Cucchietti, Sonia Fernandez Vidal and Juan Pablo Paz, Phys. Rev. A 75,032337 (2007).
[133]C.A. Sackett et al., Nature (London),404,256 (2000).
[134]H. Q. Zhou, R. Orus, G. Vidal, cond-mat/0709.4596.
[135]T. Prosen, T. H. Seligman, and M. Znidaric, Prog. Theor. Phys. Suppl. 150,200 (2003).
[136]P. Zanardi, D.A. Lidar, Phys. Rev. A 70 (2004) 012315.
[137]C. Dankert, Efficient simulation of random quantum states and operators, Math. Thesis, University of Waterloo, quant-ph/0512217.
[138]A. Ambainis and J. Emerson, quant-ph/0701126.
[139]L. H. Pedersen, N. M. Mφller, K. Mφlmer, Phys. Lett. A 367,47 (2007).
[140]C. K. Majumdar and D. K. Ghosh, J. Math. Phys.10,1388 (1969); J. Phys. C 3,911 (1970).
[141]S. Chen, L. Wang, S. J. Gu, and Y. P. Wang, Phys. Rev. E 76,061108 (2007).
[142]A. Uhlmann, Rep. Math. Phys.9,273 (1976); R. Jozsa, J. Mod. Opt.41, 2315 (1994).
[143]W. Dur, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett, 87,137901 (2001); M. S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A,67,012306 (2003).
[144]P. Zanardi, and C. Z. L. Faoro, Phys. Rev. A 62030301 (2000); P. Zanardi, Phys. Rev. A 63040304 (2001);X. Wang, and B. C. Sanders, D. W. Berry, Phys. Rev. A 67042323 (2003).
[145]Z. H. Ma and X. Wang, Phys. Rev. A 75,014304 (2007).
[146]F.C. Alcaraz and A.L. Malvezzi, J. Phys. A 30,767 (1997). T. Ono, T. Nishimura, M. Katsumua, T. Morita, and M. Sugimoto, J. Phys. Soc. Jpn. 66,2576 (1997).
[147]Guang-Shan Tian and Hai-Qing Lin, Phys. Rev. B 66,224408 (2002).
[148]Zhe Sun, Xiaoguang Wang. AnZi Hu and You-Quan Li, Physica A 370 (2006) 483-500.
[149]Shi-Jian Gu, Hai-Qing Lin, and You-Quan Li, Phys. Rev. A 68,042330 (2003).
[150]Xiaoguang Wang and Shi-Jian Gu, J. Phys. A:Math. Theor.40 (2007) 10759-10767.