量子系统中的熵与纠缠
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摘要
近年来，量子熵理论在量子物理特别是在量子光学和量子信息学领域中的广泛应用，受到人们的极大关注。它是理解和研究量子计算、量子通信、量子隐形传态、量子测量、量子纠缠等热点课题的理论基础和有力工具。光场和原子的压缩以其具有重要的应用价值引起了人们的广泛研究。最近文献[16]提出的信息熵测不准关系为高灵敏量度光场与原子的压缩效应提供了新的理论根据。
本文运用量子熵理论着重研究了量子系统中的纠缠，此外还探讨了量子力学通道、二粒子隐形传态的熵描述及原子信息熵压缩等问题。具体内容如下：
第一章简要地介绍了有关量子熵与纠缠的基本概念和理论。第二、四、五章运用量子熵理论研究了量子系统中的熵动力学性质与量子纠缠性质。其中第二章研究了双模压缩真空态在特定环境(原子)下的退相干和纠缠。这里不但讨论了原子和双模压缩真空态的纠缠，而且讨论了双模缩真空态在与原子作用过程中两个模间的纠缠。通过讨论模间纠缠和保真度，发现了双模压缩真空态在原子作用过程中仍保持较高的纠缠度和保真度。本章的工作对量子计算和量子通讯有重要的意义；第四章研究了两维囚禁离子系统中量子纠缠，提出了两束激光和囚禁离子相互作用的理论模型，利用该模型的哈密顿量既讨论了系统内外自由度(离子和振动声子)间的纠缠性质，又创新地用量子相对熵研究了振动声子两个自由度间的纠缠。研究发现：尽管振动声子两个自由度间没有发生相互作用，但由于离子和振动声子的作用，从而导致了两个自由度间纠缠的改变，并且首次从理论上揭示了离子-振动声子纠缠与振动声子两个自由度间纠缠的关系。此外还讨论了系统的初态对纠缠的影响；第五章研究了在双光子非线性J-C模型中非线性作用对原子-双模场及场模间纠缠的影响，探讨了原子崩塌-回复和纠缠之间的关系，研究结果显示：当类Kerr介
质不存在时，结果类似第四章的结果，原子和光场的纠缠及双模场之间的纠缠
没有周期性.然而当类Kerr介质存在时纠缠呈现出周期性.本章中的另二个
新发现是:在原子崩塌和回复时间区域内对应着特定的纠缠.第三章将Ohya
的互嫡理论引入到原子和双模压缩态光场的双光子相互作用过程的研究，给
出了原子互嫡的表达式，发现原子量子力学通道具有周期开关特性，并讨论了
压缩因子对量子力学通道的影响.第六章提出了实现量子信息嫡压缩的一种
新方案.该方案容易在实验上实现，是实现原子压缩的一条有效途径.第七章
给出了如何对两个二能级粒子的任意纯态进行量子隐形传送，并用嫡理论对
此作了解释.第八章是本文的总结和对有关问题的展望.
Entropy and entanglement in the quantum system
Wang Cheng-Zhi Directed by professor Fang Mao-Fa .
Department of Physics, Hunan Normal University, Changsha, 410081, China
Recently, considerable attention has been paid to the application of quantum entropy theory in quantum physics, specially, in quantum optics and quantum information. Quantum entropy theory is the foundation and powerful tool in understanding and studying such problems as quantum computation, quantum communication, quantum teleportation, quantum measurement and quantum entanglement. The squeezing of the field and atom intensively studied due to its practical application, the reference[16] presented the squeezing theory of quantum information for a high sensitive measurement of the squeezing effects of the field and atom.
In this thesis, quantum entropy theories are applied to study quantum entanglement and other problems: quantum channel ?two-particle teleportation and its interpretation in terms of entropy ?quantum information squeezing of atom and so on in quantum systems. The thesis is structured as follows:
In the first chapter, quantum entropy basal conception and related theories are concisely introduced. In the chapter 2,4 and 5, the entropic dynamics and the properties of entanglement are investigated in quantum system.' In the 2 of these chapters, the decoherence and entanglement of the two-mode squeezing vacuum state are studied under the influence of the special environment (atom). Not
only the quantum entanglement between the atom and two-mode squeezing vacuum state is studied, but also the quantum entanglement between two modes of the two-mode light field state is studied. By the studying of quantum entanglement between two modes of the two-mode light field state and fidelity of the two-mode light field state, we find that, if the initial conditions of the system are properly selected, the two-mode squeezing vacuum state used in quantum communication still keeps high degree of entanglement during the interaction between atom and light field, which is significative to quantum communication; In the chapter 4, quantum entanglement in a two-dimensional ion trap is investigated. In this chapter, the model of between two leasers and a trapped ion is presented. The entanglement between two degrees of freedom is initially investigated by using quantum relative entropy, in addition to the entanglement between two degrees of freedom(ion and phonon). The findings give that entanglement between two degrees of freedom of the phonon varies with time, although there is no interaction between the two degrees of freedom, and the relation between the two entanglements is discovered for the first time. Moreover, the findings also shows that system initial state entanglement has vast influence on the two entangiements;In chapter 5, the influence of nolinear interaction on entanglement and the relation between atomic collapse-revive and entanglement are discussed in two-mode nolinear J-C model. The first result gives that when the Kerr medium does not exist, the findings is the same as these in the chapter 4, but when the Kerr medium exists, evolutions of entanglements are periodic, another new result is that time when atom is collapsing and reviving corresponds to the special entanglement. Ohya mutual entropic theory is introduced into the study on the two-photon process of atom and two-mode squeezing vacuum state in chapter 3. The expression of atomic mutual entropy is derived. Periodicity of. the quantum mechanical channel is displayed , and the influence of squeezing coefficient on atomic mutual entropy is investigated; chapter 6 gives a scheme for realizing the squeezing
of atomic information entropy is brought forward . This scheme is easily experimentally performed and is a effective avenue to realize the atomic squeezing; In chapter 7, a scheme for quantum teleportation of two-particle arbitrary pure two-levelstates is proposed, whose interpretation in terms of quantum information is given by using quantum entropy t
本文运用量子熵理论着重研究了量子系统中的纠缠，此外还探讨了量子力学通道、二粒子隐形传态的熵描述及原子信息熵压缩等问题。具体内容如下：
第一章简要地介绍了有关量子熵与纠缠的基本概念和理论。第二、四、五章运用量子熵理论研究了量子系统中的熵动力学性质与量子纠缠性质。其中第二章研究了双模压缩真空态在特定环境(原子)下的退相干和纠缠。这里不但讨论了原子和双模压缩真空态的纠缠，而且讨论了双模缩真空态在与原子作用过程中两个模间的纠缠。通过讨论模间纠缠和保真度，发现了双模压缩真空态在原子作用过程中仍保持较高的纠缠度和保真度。本章的工作对量子计算和量子通讯有重要的意义；第四章研究了两维囚禁离子系统中量子纠缠，提出了两束激光和囚禁离子相互作用的理论模型，利用该模型的哈密顿量既讨论了系统内外自由度(离子和振动声子)间的纠缠性质，又创新地用量子相对熵研究了振动声子两个自由度间的纠缠。研究发现：尽管振动声子两个自由度间没有发生相互作用，但由于离子和振动声子的作用，从而导致了两个自由度间纠缠的改变，并且首次从理论上揭示了离子-振动声子纠缠与振动声子两个自由度间纠缠的关系。此外还讨论了系统的初态对纠缠的影响；第五章研究了在双光子非线性J-C模型中非线性作用对原子-双模场及场模间纠缠的影响，探讨了原子崩塌-回复和纠缠之间的关系，研究结果显示：当类Kerr介
质不存在时，结果类似第四章的结果，原子和光场的纠缠及双模场之间的纠缠
没有周期性.然而当类Kerr介质存在时纠缠呈现出周期性.本章中的另二个
新发现是:在原子崩塌和回复时间区域内对应着特定的纠缠.第三章将Ohya
的互嫡理论引入到原子和双模压缩态光场的双光子相互作用过程的研究，给
出了原子互嫡的表达式，发现原子量子力学通道具有周期开关特性，并讨论了
压缩因子对量子力学通道的影响.第六章提出了实现量子信息嫡压缩的一种
新方案.该方案容易在实验上实现，是实现原子压缩的一条有效途径.第七章
给出了如何对两个二能级粒子的任意纯态进行量子隐形传送，并用嫡理论对
此作了解释.第八章是本文的总结和对有关问题的展望.
Entropy and entanglement in the quantum system
Wang Cheng-Zhi Directed by professor Fang Mao-Fa .
Department of Physics, Hunan Normal University, Changsha, 410081, China
Recently, considerable attention has been paid to the application of quantum entropy theory in quantum physics, specially, in quantum optics and quantum information. Quantum entropy theory is the foundation and powerful tool in understanding and studying such problems as quantum computation, quantum communication, quantum teleportation, quantum measurement and quantum entanglement. The squeezing of the field and atom intensively studied due to its practical application, the reference[16] presented the squeezing theory of quantum information for a high sensitive measurement of the squeezing effects of the field and atom.
In this thesis, quantum entropy theories are applied to study quantum entanglement and other problems: quantum channel ?two-particle teleportation and its interpretation in terms of entropy ?quantum information squeezing of atom and so on in quantum systems. The thesis is structured as follows:
In the first chapter, quantum entropy basal conception and related theories are concisely introduced. In the chapter 2,4 and 5, the entropic dynamics and the properties of entanglement are investigated in quantum system.' In the 2 of these chapters, the decoherence and entanglement of the two-mode squeezing vacuum state are studied under the influence of the special environment (atom). Not
only the quantum entanglement between the atom and two-mode squeezing vacuum state is studied, but also the quantum entanglement between two modes of the two-mode light field state is studied. By the studying of quantum entanglement between two modes of the two-mode light field state and fidelity of the two-mode light field state, we find that, if the initial conditions of the system are properly selected, the two-mode squeezing vacuum state used in quantum communication still keeps high degree of entanglement during the interaction between atom and light field, which is significative to quantum communication; In the chapter 4, quantum entanglement in a two-dimensional ion trap is investigated. In this chapter, the model of between two leasers and a trapped ion is presented. The entanglement between two degrees of freedom is initially investigated by using quantum relative entropy, in addition to the entanglement between two degrees of freedom(ion and phonon). The findings give that entanglement between two degrees of freedom of the phonon varies with time, although there is no interaction between the two degrees of freedom, and the relation between the two entanglements is discovered for the first time. Moreover, the findings also shows that system initial state entanglement has vast influence on the two entangiements;In chapter 5, the influence of nolinear interaction on entanglement and the relation between atomic collapse-revive and entanglement are discussed in two-mode nolinear J-C model. The first result gives that when the Kerr medium does not exist, the findings is the same as these in the chapter 4, but when the Kerr medium exists, evolutions of entanglements are periodic, another new result is that time when atom is collapsing and reviving corresponds to the special entanglement. Ohya mutual entropic theory is introduced into the study on the two-photon process of atom and two-mode squeezing vacuum state in chapter 3. The expression of atomic mutual entropy is derived. Periodicity of. the quantum mechanical channel is displayed , and the influence of squeezing coefficient on atomic mutual entropy is investigated; chapter 6 gives a scheme for realizing the squeezing
of atomic information entropy is brought forward . This scheme is easily experimentally performed and is a effective avenue to realize the atomic squeezing; In chapter 7, a scheme for quantum teleportation of two-particle arbitrary pure two-levelstates is proposed, whose interpretation in terms of quantum information is given by using quantum entropy t
引文
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[28] E. T. Jaynes, F. W. Cummings, Proc, IEEE, 51(1967), 89.
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[31] S. J. D. Phoenix, P. L. Knight, Ann. Phys., 186(1998), 381.
[32] E. Schmidt Math. Ann. 63(1907), 433.
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[36] D. Bouwmeester et al, Phys. Rev. Lett., 82(1999), 1345.
[37] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. knight, Phys. Rev. Lett., 78(1997), 2275.
[38] V. Vedral, M. B. Plenio, Phys. Rev., A57(1998), 1619.
[39] E. Rains, Phys. Rev., A60(1999),179.
[40] S. J. Wu and Y. D. Zhang, LANL e-print Quant-ph/O004018.
[41] C. H. Bennett, S. Popescu, B. Schumacher, J. A. Somlin, W. K. Wootters, Phys. Rev. Lett 76(1996),722.
[42] 喀兴林，高等量子力学，高等教育出版社，北京(2000)．
[43] Cirac J I and Zoller P 1995 Phys. Rev. Lett. 74 4091.
[44] Meekhof D M, Monroe C, King B E, Itano W M and Wineland D J 1996 Phys. Rev. lett.76 1796; C. Moaroe, Meekhof D M, King B E, Wineland D J 1996 Science 212 1131; Gerry C C, Gou S-C, and Steinbach J 1997 Phys. Rev. A 55 630.
[45] Fang M F, Swain S and Zhou P 2001 Phys. Rev. A 63 013812.
[46] Sukumar C V and Buck B 1981 Phys. Lett. A83 211.
[47] Gerry C C, Gou S-C, and Steinbach J 997 Phys. Rev. A55 630.
[48] Hou L Z and Zeng G J 2001 Chin. Phys. 10 821.
[49] Wu S J and Zhang Y D, LANL e-print Quant-ph/0004018; Rains E 1999 Phys. Rev. A60 179.
[50] B. W. Shore, and P. L. Knight, J. Mod. OPt, 40(1993),1195.
[51] J. H. Eberly, N. B. Narozhny, and Sanchez-Mondragon, Phys. Rev. Lett., 44(1980),1323.
[52] Phoenix, and P. L. Knight, Ann. Phys.,186(1991),381.
[53] Li Gao Xing, Peng Jin Sheng, Zhou Peng, Phys. Lett., 12(1995),79.
[54] L. Vestergaard Hau, et al, Nature, 397(1999), 594; M. M. Kash, et al, Phys. Rev. Lett 82,5229.
[55] M. Genovese, and C. Novero, Phys. Lett., 85(2000),445.
[56] H. I. Yoo and J. H. Eberly, Phys. Rep., 118(1985),239.
[57] S. C. Gou, Phys. Rev. A 40(1989)5116.
[58] J. Aliaga, and J. L. Gruver, Phys. Lett. A 221(1996),19.
[59] Guido Berlin and J. aliaga, J. Mod. Opt 48(2001),1819.
[60] D. J. Wineland and J. J. Bollinger, Phys. Rev. A, 50(1994), 67.
[61] J. L. Srensen and J. Hald, Phys. Rev. Lett., 80(1998), 3487.
[62] L. Kuang, Commun. Theor. Phys., 30(1998), 161; M. G. Moore and P. Meystre, Phys. Rev. A, 59(1999), R1754.
[63] D. Stoler, Phys. Rev. D, 1(1970), 3217; K. Wodkiewicz, Phys. Rev. B, 32(1994), 4750.
[64] D. F. Walls and P. Zoller, Phys. Rev. Lett., 47(1981), 709.
[65] Peng J S, Li G X, Introduction To Quantum Optics, World Scientic Press(1998)
[66] Stefano Mancini and Rodolfo Bonifacio, Phys. Rev. A 63(2001),032310.
[67] D. Boachi, S. Branca, F. G. Martini, L. Hardy, and S. Popescu Phys. Rev. Lett 80(1998), 1121; A. Furusawa, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik Science 282(1998), 706.
[68] Stenholm S and Bardroff P J Phys. Rev. A 68 (1998), 4373.
[69] S. L. Braunstein, H. J. Kimble Phys. Rev. Lett 80 (1998), 869; S. L. Braunstein Nature 394(1998),47.
[70] J. Lee and M. S. Kim Phys. Rev. Lett 84(2000), 4236.
[71] M. Ikrm, Zhu S Y, and M. S. Zubairy Phys. Rev. A 62(2000), 022307.
[72] Ye L, Yao C M and Guo G C Chin Phys 10(2001),1003
[26] L. K. Grover, Phys. Rev. Lett., 79(1997), 325.
[27] A. Furusawa, J. L. SOrensen, S. L. Braunstein, et al, Science, 282(1998), 706.
[28] E. T. Jaynes, F. W. Cummings, Proc, IEEE, 51(1967), 89.
[29] E. B. Davics, Quantum Theory Of Open System(Academic Press, New York(1976), 241).
[30] E. Solano, M. F. Santos, and P. Milman, Phys. Rew. A64(2001), 024304.
[31] S. J. D. Phoenix, P. L. Knight, Ann. Phys., 186(1998), 381.
[32] E. Schmidt Math. Ann. 63(1907), 433.
[33] X. Liu, M. F. Fang and A. L. Liu, Acta. Physica, 49(2000), 1707(in Chinese)[刘翔，方卯发，刘安玲，物理学，49(2000)，1707].
[34] S. M. Ao, S. L. Zhou and G. J. Zeng, Acta. Physica, 50(2001), 52 (in Chinese)[敖胜美，周石伦和曾高坚，物理学报，50(2001),52.
[35] H. Araki and E. Lieb, Commun. Math. Phys., 18(1970), 160.
[36] D. Bouwmeester et al, Phys. Rev. Lett., 82(1999), 1345.
[37] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. knight, Phys. Rev. Lett., 78(1997), 2275.
[38] V. Vedral, M. B. Plenio, Phys. Rev., A57(1998), 1619.
[39] E. Rains, Phys. Rev., A60(1999),179.
[40] S. J. Wu and Y. D. Zhang, LANL e-print Quant-ph/O004018.
[41] C. H. Bennett, S. Popescu, B. Schumacher, J. A. Somlin, W. K. Wootters, Phys. Rev. Lett 76(1996),722.
[42] 喀兴林，高等量子力学，高等教育出版社，北京(2000)．
[43] Cirac J I and Zoller P 1995 Phys. Rev. Lett. 74 4091.
[44] Meekhof D M, Monroe C, King B E, Itano W M and Wineland D J 1996 Phys. Rev. lett.76 1796; C. Moaroe, Meekhof D M, King B E, Wineland D J 1996 Science 212 1131; Gerry C C, Gou S-C, and Steinbach J 1997 Phys. Rev. A 55 630.
[45] Fang M F, Swain S and Zhou P 2001 Phys. Rev. A 63 013812.
[46] Sukumar C V and Buck B 1981 Phys. Lett. A83 211.
[47] Gerry C C, Gou S-C, and Steinbach J 997 Phys. Rev. A55 630.
[48] Hou L Z and Zeng G J 2001 Chin. Phys. 10 821.
[49] Wu S J and Zhang Y D, LANL e-print Quant-ph/0004018; Rains E 1999 Phys. Rev. A60 179.
[50] B. W. Shore, and P. L. Knight, J. Mod. OPt, 40(1993),1195.
[51] J. H. Eberly, N. B. Narozhny, and Sanchez-Mondragon, Phys. Rev. Lett., 44(1980),1323.
[52] Phoenix, and P. L. Knight, Ann. Phys.,186(1991),381.
[53] Li Gao Xing, Peng Jin Sheng, Zhou Peng, Phys. Lett., 12(1995),79.
[54] L. Vestergaard Hau, et al, Nature, 397(1999), 594; M. M. Kash, et al, Phys. Rev. Lett 82,5229.
[55] M. Genovese, and C. Novero, Phys. Lett., 85(2000),445.
[56] H. I. Yoo and J. H. Eberly, Phys. Rep., 118(1985),239.
[57] S. C. Gou, Phys. Rev. A 40(1989)5116.
[58] J. Aliaga, and J. L. Gruver, Phys. Lett. A 221(1996),19.
[59] Guido Berlin and J. aliaga, J. Mod. Opt 48(2001),1819.
[60] D. J. Wineland and J. J. Bollinger, Phys. Rev. A, 50(1994), 67.
[61] J. L. Srensen and J. Hald, Phys. Rev. Lett., 80(1998), 3487.
[62] L. Kuang, Commun. Theor. Phys., 30(1998), 161; M. G. Moore and P. Meystre, Phys. Rev. A, 59(1999), R1754.
[63] D. Stoler, Phys. Rev. D, 1(1970), 3217; K. Wodkiewicz, Phys. Rev. B, 32(1994), 4750.
[64] D. F. Walls and P. Zoller, Phys. Rev. Lett., 47(1981), 709.
[65] Peng J S, Li G X, Introduction To Quantum Optics, World Scientic Press(1998)
[66] Stefano Mancini and Rodolfo Bonifacio, Phys. Rev. A 63(2001),032310.
[67] D. Boachi, S. Branca, F. G. Martini, L. Hardy, and S. Popescu Phys. Rev. Lett 80(1998), 1121; A. Furusawa, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik Science 282(1998), 706.
[68] Stenholm S and Bardroff P J Phys. Rev. A 68 (1998), 4373.
[69] S. L. Braunstein, H. J. Kimble Phys. Rev. Lett 80 (1998), 869; S. L. Braunstein Nature 394(1998),47.
[70] J. Lee and M. S. Kim Phys. Rev. Lett 84(2000), 4236.
[71] M. Ikrm, Zhu S Y, and M. S. Zubairy Phys. Rev. A 62(2000), 022307.
[72] Ye L, Yao C M and Guo G C Chin Phys 10(2001),1003