连续变量纠缠态的量子非局域性
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摘要
量子非局域性是量子理论中令人感到奇妙的特征之一。尽管和经典理论一样,量子理论(从非相对论量子力学到相对论量子场论)采用了局域描述的方法,但就本质而言,量子理论依然是在局域描述外衣下的空间非局域理论。非局域性这一奇妙的性质正越来越多的展现出来,并且正在经受着越来越多的实验检验。研究量子非局域性对量子理论的根本问题和量子信息论是有意义的。
全文分为六章。第一章简要介绍研究量子非局域性的历史和一些重要成果。
第二章介绍量子非局域性的基本理论;包括,EPR佯谬和量子力学的完备性;Bell不等式及其破坏;扩展的CHSH不等式及其最大破坏。重点介绍的是带不等式的Bell型定理。
第三章研究纠缠相干态的量子非局域性;纠缠相干态是少数几种实验上可以制备的连续变量纠缠态之一。研究它不仅可以对连续变量纠缠态的非经典性质作深入了解,同时也能对多个关联光子之间的非局域性质有所探究。本章首先介绍了计算纠缠相干态量子非局域性的两种通用方法。接着提出一种全新的方法,即,基于宇称相干态构成的膺自旋算符可以使Bell-CHSH不等式的违背值达到它的上限2~(1/2),同时在实验上具有实现的可能性。之后找到了在理论上使纠缠相干态获得最优非局域性的Bell算符。
第四章研究两模减k个光子压缩真空态的量子非局域性;两模减k个光子压缩真空态是另一种实验上可以制备的连续变量纠缠态。它是非高斯型的量子态,可以用于非局域性的实验检验中。本章重点研究了利用膺自旋算符作用于两模减k个光子压缩真空态的Bell-CHSH不等式,发现减光子的手段可以增强该态的非局域性。然后进一步研究了退相干对该态的Bell-CHSH不等式的影响。
第五章介绍无不等式的Bell型量子非局域性定理;前四章介绍的都是带不等式的Bell型定理。这一章沿着无不等式的GHZ,Hardy和Cabello定理做了一个综述,然后提出一个新的连续变量的Cabello定
Quantum nonlocality is one of the amazing properties in quantum theory. Although, quantum theory (from nonrelative quantum mechanics to relative quantum field theories) describes the world in terms of locality, like the classic theory. But as far as its essence concerned, quantum theory is still a space nonlocal theory in terms of locality. The nonlocal property has been displayed. There are more and more experiments to test the nonlocality. Studying the quantum nonlocality is of great importance to the basic problems in quantum theorems and quantum information theory.This thesis is composed of six chapter. The first chapter is aimed to introduce the history and some main fruits of quantum nonlocality.The second part introduces the basic theories of quantum nonlocality including EPR paradox and the completeness of quantum mechanics, the Bell inequality and its violation and the generalized CHSH inequality and its maximal violation. Then emphasis of this part is to introduce the Bell-type theorem with inequalties.The third chapter investigates the quantum nonlocaity of entangled coherent states, which is an important continuous-variable entangled states and can be generated in experiment. This part introduces two general methods to calculate the quantum nonlocality of entangled coherent states. Then we propose a new method to make the Bell-CHSH inequality reach the maximal violation value 2(2~(1/2)). This method has the probability to realize in experiment. Then we find the Bell operator which can make the entangled coherent states get the optimal quantum nonlocality theoretically.The fourth chapter studies the quantum nonlocality of the k-photon subtraction squeezed vacuum states, which is another continuous-variable entangled states and can be generated in experiments. It is a non-Gaussian quantum state and can be used to test the nonlocality experimently. This chapter emphasizes on the Bell-CHSH inequality of the two-mode k-photon subtraction
vacuum states in terms of pseudospin operators. We find that the subtraction of photons can enhance the quantum nonlocality of these states. Then we further study the effect the decoherence has on the Bell-CHSH inequality.The fifth chapter introduces the Bell-type quantum nolocal theorem without inequalities. This chapter make a genenralization of GHZ, Hardy and Ca-bello theorem without inequality. Then we propose a new continuous-variable Cabello theorem, which used super-entangled states with parity and polarization freedoms and give a method without inequality to test quantum nonlocality. This is an All-Versus-Nothing method.The second section of the third chapter, the second and the third sections of the fourth chapter and the second section of the fifth chapter are inivations.In the sixth chapter we give a concise conclusion of this thesis and make an expectation of the future in this field.
全文分为六章。第一章简要介绍研究量子非局域性的历史和一些重要成果。
第二章介绍量子非局域性的基本理论;包括,EPR佯谬和量子力学的完备性;Bell不等式及其破坏;扩展的CHSH不等式及其最大破坏。重点介绍的是带不等式的Bell型定理。
第三章研究纠缠相干态的量子非局域性;纠缠相干态是少数几种实验上可以制备的连续变量纠缠态之一。研究它不仅可以对连续变量纠缠态的非经典性质作深入了解,同时也能对多个关联光子之间的非局域性质有所探究。本章首先介绍了计算纠缠相干态量子非局域性的两种通用方法。接着提出一种全新的方法,即,基于宇称相干态构成的膺自旋算符可以使Bell-CHSH不等式的违背值达到它的上限2~(1/2),同时在实验上具有实现的可能性。之后找到了在理论上使纠缠相干态获得最优非局域性的Bell算符。
第四章研究两模减k个光子压缩真空态的量子非局域性;两模减k个光子压缩真空态是另一种实验上可以制备的连续变量纠缠态。它是非高斯型的量子态,可以用于非局域性的实验检验中。本章重点研究了利用膺自旋算符作用于两模减k个光子压缩真空态的Bell-CHSH不等式,发现减光子的手段可以增强该态的非局域性。然后进一步研究了退相干对该态的Bell-CHSH不等式的影响。
第五章介绍无不等式的Bell型量子非局域性定理;前四章介绍的都是带不等式的Bell型定理。这一章沿着无不等式的GHZ,Hardy和Cabello定理做了一个综述,然后提出一个新的连续变量的Cabello定
Quantum nonlocality is one of the amazing properties in quantum theory. Although, quantum theory (from nonrelative quantum mechanics to relative quantum field theories) describes the world in terms of locality, like the classic theory. But as far as its essence concerned, quantum theory is still a space nonlocal theory in terms of locality. The nonlocal property has been displayed. There are more and more experiments to test the nonlocality. Studying the quantum nonlocality is of great importance to the basic problems in quantum theorems and quantum information theory.This thesis is composed of six chapter. The first chapter is aimed to introduce the history and some main fruits of quantum nonlocality.The second part introduces the basic theories of quantum nonlocality including EPR paradox and the completeness of quantum mechanics, the Bell inequality and its violation and the generalized CHSH inequality and its maximal violation. Then emphasis of this part is to introduce the Bell-type theorem with inequalties.The third chapter investigates the quantum nonlocaity of entangled coherent states, which is an important continuous-variable entangled states and can be generated in experiment. This part introduces two general methods to calculate the quantum nonlocality of entangled coherent states. Then we propose a new method to make the Bell-CHSH inequality reach the maximal violation value 2(2~(1/2)). This method has the probability to realize in experiment. Then we find the Bell operator which can make the entangled coherent states get the optimal quantum nonlocality theoretically.The fourth chapter studies the quantum nonlocality of the k-photon subtraction squeezed vacuum states, which is another continuous-variable entangled states and can be generated in experiments. It is a non-Gaussian quantum state and can be used to test the nonlocality experimently. This chapter emphasizes on the Bell-CHSH inequality of the two-mode k-photon subtraction
vacuum states in terms of pseudospin operators. We find that the subtraction of photons can enhance the quantum nonlocality of these states. Then we further study the effect the decoherence has on the Bell-CHSH inequality.The fifth chapter introduces the Bell-type quantum nolocal theorem without inequalities. This chapter make a genenralization of GHZ, Hardy and Ca-bello theorem without inequality. Then we propose a new continuous-variable Cabello theorem, which used super-entangled states with parity and polarization freedoms and give a method without inequality to test quantum nonlocality. This is an All-Versus-Nothing method.The second section of the third chapter, the second and the third sections of the fourth chapter and the second section of the fifth chapter are inivations.In the sixth chapter we give a concise conclusion of this thesis and make an expectation of the future in this field.
引文
[1] M. Jammer. The Philosophy of Quantum Mechanics [M]. New York: Wiley-Interscience Publications, 2000, pp. 115-116.
[2] 张永德.量子信息物理原理[M].北京:科学出版社,2006.
[3] A. Einstein, B. Podolsky and N. Rosen. Can Quantum-Mechanical De-scription of Physical Reality Be Considered Complete[J]? Phys. Rev., 1935, 47(10): 777-780.
[4] 候德彭量子理论[M].北京:商务印书馆,1982.
[5] J. S. Bell. On the Einstein Poldolsky Rosen paradox [J]. Physics, 1964, 1: 195.
[6] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt. Proposed experiment to test local hidden-variable theories [J]. Phys. Rev. Lett., 1969, 23: 880-884.
[7] D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger. Bell's theorem without inequalities [J]. Am. J. Phys., 1990, 58: 1131.
[8] S.M. Tan, D.F. Walls, and M.J. Collett. Nonlocality of a single photon [J]. Phys. Rev. Lett., 1991, 66: 252.
[9] P. Grangier, G. Roger, and A. Aspect. Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences [J]. Europhys. Lett., 1986, 1: 173.
[10] L. Hardy. Nonlocality to two particles without inequalities for almost all entangled states [J]. Phys. Rev. Lett., 1993, 71: 1665.
[11] S. Goldstein. Nonlocality without inequalities for almost all entangled states for two particles [J]. Phys. Rev. Lett., 1994, 72: 1951.
[12] L. Hardy. Nonlocality of a single photon revisited [J]. Phys. Rev. Lett., 1994, 73: 2279.
[13] G. Weihs, et.al.. Violation of Bell's Inequality under Strict Einstein Locality Conditions [J]. Phys. Rev. Lett., 1998, 81: 5039.
[14] J. W. Pan, et.al.. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement [J]. Nature, 2000, 403: 515.
[15] A. Cabello. Bell's Theorem without Inequalities and without Proba-bilities for Two Observers [J]. Phys. Rev. Lett., 2001, 86: 1911.
[16] A. Cabello. " All versus Nothing" Inseparability for Two Observers [J]. Phys. Rev. Lett., 2001, 87: 010403.
[17] Z.B. Chen, J. W. Pan, G. Hou and Y. D. Zhang. Maximal Violation of Bell's Inequalities for Continuous Variable Systems [J]. Phys. Rev. Lett., 2002, 88: 040406.
[18] Z. B. Chen, J. W. Pan, Y. D. Zhang, C. Brukner and A. Zeilinger. All-Versus-Nothing Violation of Local Realism for Two Entangled Photons [J]. Phys. Rev. Lett., 2003, 90: 160408.
[19] B. Hessmo, et.al.. Experimental Demonstration of Single Photon Non-locality [J]. Phys. Rev. Lett., 2004, 92: 180401.
[20] T. Yang, et. al.. All-Versus-Nothing Violation of Local Realism by Two-Photon, Four-Dimensional Entanglement [J]. Phys. Rev. Lett., 2005, 95: 240406.
[21] 张永德.量子力学[M].北京:科学出版社,2003.
[22] S. L. Braustein, A. Mann and M. Revzen. Maximal violation of Bell in-equalities for mixed states [J]. Phys. Rev. Lett., 1992, 68: 3259.
[23] B.C. Sanders. Entangled coherent states [J]. Phys. Rev. A, 1992, 45: 6811.
[24] D. Wilson, H. Jeong and M. S. Kim Quantum Nonlocality for a Mixed Entangled Coherent State [J]. J. Mod. Opt., 2002, 49: 851.
[25] A. Garg and D. Mermin. Detector inefficiencies in the Einstein-Podolsky-Rosen experiment [J]. Phys. Rev. D, 1987, 35: 3831.
[26] J. S. Bell. Speakable and Unspeakable in Quantum Mechanics [M]. Cambridge: Cambridge University Press, 1987, Chap. 21.
[27] K. Banaszek and K. Wodkiewicz. Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation. [J] Phys. Rev. A, 1998, 58: 4345.
[28] H. Jeong. Quantum nonlocality test for continuous-variable states with dichotomic observables [J]. Phys. Rev. A, 2003, 67: 012106.
[29] C. Monroe, D.M. Meekhof, B.E. King, and D.J. Wineland. A "Schrodinger Cat" Superposition State of an Atom [J]. Science, 1996, 272: 1131.
[30] M. Brune, E. Hagley, J. Dreyer, X. Maitre, C. Wunderlich, J.M. Raimond, and S. Haroche. Observing the Progressive Decoherence of the " Meter " in a Quantum Measurement [J]. Phys. Rev. Lett., 1996, 77: 4887.
[31] J.M. Raimond J M, M. Brune and S. Haroche. Reversible Decoherence of a Mesoscopic Superposition of Field States [J]. Phys. Rev. Lett., 1997, 79: 1964.
[32] H. Jeong, M.S. Kim, T.C. Ralph, and B.S. Ham. Generation of macroscopic superposition states with small nonlinearity [J]. Phys. Rev. A, 2004, 70: 061801(R).
[33] A.P. Lund, H. Jeong, T.C. Ralph, and M.S. Kim. Conditional production of superpositions of coherent states with inefficient photon detection [J]. Phys. Rev. A, 2004, 70: 020101(R).
[34] H. Jeong, A.P. Lund, and T.C. Ralph. Production of superpositions of coherent states in traveling optical fields with inefficient photon de- tection [J]. Phys. Rev. A, 2005, 72: 013801.
[35] S.J. Van Enk and O. Hirota. Entangled coherent states: Teleportation and decoherence [J]. Phys. Rev. A, 2001, 64: 022313.
[36] C.L. Chai. Two-mode nonclassical state via superpositions of two-mode coherent states [J]. Phys. Rev. A, 1992, 46: 7187.
[37] S.L. Braunstein, H.J. Kimble. Teleportation of Continuous Quantum Variables [J]. Phys. Rev. Lett., 1998, 80: 869.
[38] J. Wenger, R. Tualle-Brouri and PH. Grangier. Non-Gaussian Statistics from Individual Pulses of Squeezed Light [J]. Phys. Rev. Lett., 2004, 92: 153601.
[39] R. Garcia-Patron, et.al.. Proposal for a Loophole-Free Bell Test Using Homodyne Detection [J]. Phys. Rev. Lett., 2004, 93: 130409.
[40] R. Garcia-Patron, J. Fiurasek, N. J. Cerf. Loophole-free test of quantum nonlocality using high-efficiency homodyne detectors [J]. Phys. Rev. A, 2005, 71: 022105.
[41] J. W. Pan. Quantum teleportation and Multi-photon Entanglement, the dissertation for PhD [M]. Institute for Experimental Physics, University of Vienna, 1998.
[42] P. T. Cochrane, T. C. Ralph and G. J. Milburn. Teleportation improvement by conditional measurements on the two-mode squeezed vacuum [J]. Phys. Rev. A, 2002, 65: 062306.
[43] T. Opatrny, G. Kurizki and D. G. Welsch. Improvement on teleportation of continuous variables by photon subtraction via conditional measurement [J]. Phys. Rev. A, 2000, 61: 032302.
[44] A. Kitagawa, M. Takeoka, K. Wakui and M. Sasaki. Quantum gloves: Quantum states that encode as much as possible chirality and nothing else [J]. Phys. Rev. A, 2005, 72: 022304.
[45] D. E. Browne, J. Eisert, S. Scheel and M. B. Plenio. Driving non-Gaussian to Gaussian states with linear optics [J]. Phys. Rev. A, 2003, 67: 062320.
[46] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen. Algebraic approch to master equations with superoperator generators of su.1,1. and su.2. Lie algebras [J]. Phys. Rev. A, 2003, 67: 024101.
[47] S. J. Wu and Y. D. Zhang. Calculating the relative entropy of entangle- ment [J]. Phys. Rev. A, 2004, 61: 012308.
[48] N. Lutkenhaus, J. Calsamiglia, and K.-A. Suominen. Bell measurements for teleportation [J]. Phys. Rev. A, 1999, 59: 3295.
[49] L. Vaidman, and N. Yoran. Methods for reliable teleportation [J]. Phys. Rev. A, 1999, 59: 116.
[50] N. Korolkova, et.al.. Polarization squeezing and continuous-variable polarization entanglement [J]. Phys. Rev. A, 2002, 65: 052306.
[51] C. W. Gardiner and P. Zoller Quantum Noise, [M], 2nd edn. Berlin: Springer,2000.
[2] 张永德.量子信息物理原理[M].北京:科学出版社,2006.
[3] A. Einstein, B. Podolsky and N. Rosen. Can Quantum-Mechanical De-scription of Physical Reality Be Considered Complete[J]? Phys. Rev., 1935, 47(10): 777-780.
[4] 候德彭量子理论[M].北京:商务印书馆,1982.
[5] J. S. Bell. On the Einstein Poldolsky Rosen paradox [J]. Physics, 1964, 1: 195.
[6] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt. Proposed experiment to test local hidden-variable theories [J]. Phys. Rev. Lett., 1969, 23: 880-884.
[7] D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger. Bell's theorem without inequalities [J]. Am. J. Phys., 1990, 58: 1131.
[8] S.M. Tan, D.F. Walls, and M.J. Collett. Nonlocality of a single photon [J]. Phys. Rev. Lett., 1991, 66: 252.
[9] P. Grangier, G. Roger, and A. Aspect. Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences [J]. Europhys. Lett., 1986, 1: 173.
[10] L. Hardy. Nonlocality to two particles without inequalities for almost all entangled states [J]. Phys. Rev. Lett., 1993, 71: 1665.
[11] S. Goldstein. Nonlocality without inequalities for almost all entangled states for two particles [J]. Phys. Rev. Lett., 1994, 72: 1951.
[12] L. Hardy. Nonlocality of a single photon revisited [J]. Phys. Rev. Lett., 1994, 73: 2279.
[13] G. Weihs, et.al.. Violation of Bell's Inequality under Strict Einstein Locality Conditions [J]. Phys. Rev. Lett., 1998, 81: 5039.
[14] J. W. Pan, et.al.. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement [J]. Nature, 2000, 403: 515.
[15] A. Cabello. Bell's Theorem without Inequalities and without Proba-bilities for Two Observers [J]. Phys. Rev. Lett., 2001, 86: 1911.
[16] A. Cabello. " All versus Nothing" Inseparability for Two Observers [J]. Phys. Rev. Lett., 2001, 87: 010403.
[17] Z.B. Chen, J. W. Pan, G. Hou and Y. D. Zhang. Maximal Violation of Bell's Inequalities for Continuous Variable Systems [J]. Phys. Rev. Lett., 2002, 88: 040406.
[18] Z. B. Chen, J. W. Pan, Y. D. Zhang, C. Brukner and A. Zeilinger. All-Versus-Nothing Violation of Local Realism for Two Entangled Photons [J]. Phys. Rev. Lett., 2003, 90: 160408.
[19] B. Hessmo, et.al.. Experimental Demonstration of Single Photon Non-locality [J]. Phys. Rev. Lett., 2004, 92: 180401.
[20] T. Yang, et. al.. All-Versus-Nothing Violation of Local Realism by Two-Photon, Four-Dimensional Entanglement [J]. Phys. Rev. Lett., 2005, 95: 240406.
[21] 张永德.量子力学[M].北京:科学出版社,2003.
[22] S. L. Braustein, A. Mann and M. Revzen. Maximal violation of Bell in-equalities for mixed states [J]. Phys. Rev. Lett., 1992, 68: 3259.
[23] B.C. Sanders. Entangled coherent states [J]. Phys. Rev. A, 1992, 45: 6811.
[24] D. Wilson, H. Jeong and M. S. Kim Quantum Nonlocality for a Mixed Entangled Coherent State [J]. J. Mod. Opt., 2002, 49: 851.
[25] A. Garg and D. Mermin. Detector inefficiencies in the Einstein-Podolsky-Rosen experiment [J]. Phys. Rev. D, 1987, 35: 3831.
[26] J. S. Bell. Speakable and Unspeakable in Quantum Mechanics [M]. Cambridge: Cambridge University Press, 1987, Chap. 21.
[27] K. Banaszek and K. Wodkiewicz. Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation. [J] Phys. Rev. A, 1998, 58: 4345.
[28] H. Jeong. Quantum nonlocality test for continuous-variable states with dichotomic observables [J]. Phys. Rev. A, 2003, 67: 012106.
[29] C. Monroe, D.M. Meekhof, B.E. King, and D.J. Wineland. A "Schrodinger Cat" Superposition State of an Atom [J]. Science, 1996, 272: 1131.
[30] M. Brune, E. Hagley, J. Dreyer, X. Maitre, C. Wunderlich, J.M. Raimond, and S. Haroche. Observing the Progressive Decoherence of the " Meter " in a Quantum Measurement [J]. Phys. Rev. Lett., 1996, 77: 4887.
[31] J.M. Raimond J M, M. Brune and S. Haroche. Reversible Decoherence of a Mesoscopic Superposition of Field States [J]. Phys. Rev. Lett., 1997, 79: 1964.
[32] H. Jeong, M.S. Kim, T.C. Ralph, and B.S. Ham. Generation of macroscopic superposition states with small nonlinearity [J]. Phys. Rev. A, 2004, 70: 061801(R).
[33] A.P. Lund, H. Jeong, T.C. Ralph, and M.S. Kim. Conditional production of superpositions of coherent states with inefficient photon detection [J]. Phys. Rev. A, 2004, 70: 020101(R).
[34] H. Jeong, A.P. Lund, and T.C. Ralph. Production of superpositions of coherent states in traveling optical fields with inefficient photon de- tection [J]. Phys. Rev. A, 2005, 72: 013801.
[35] S.J. Van Enk and O. Hirota. Entangled coherent states: Teleportation and decoherence [J]. Phys. Rev. A, 2001, 64: 022313.
[36] C.L. Chai. Two-mode nonclassical state via superpositions of two-mode coherent states [J]. Phys. Rev. A, 1992, 46: 7187.
[37] S.L. Braunstein, H.J. Kimble. Teleportation of Continuous Quantum Variables [J]. Phys. Rev. Lett., 1998, 80: 869.
[38] J. Wenger, R. Tualle-Brouri and PH. Grangier. Non-Gaussian Statistics from Individual Pulses of Squeezed Light [J]. Phys. Rev. Lett., 2004, 92: 153601.
[39] R. Garcia-Patron, et.al.. Proposal for a Loophole-Free Bell Test Using Homodyne Detection [J]. Phys. Rev. Lett., 2004, 93: 130409.
[40] R. Garcia-Patron, J. Fiurasek, N. J. Cerf. Loophole-free test of quantum nonlocality using high-efficiency homodyne detectors [J]. Phys. Rev. A, 2005, 71: 022105.
[41] J. W. Pan. Quantum teleportation and Multi-photon Entanglement, the dissertation for PhD [M]. Institute for Experimental Physics, University of Vienna, 1998.
[42] P. T. Cochrane, T. C. Ralph and G. J. Milburn. Teleportation improvement by conditional measurements on the two-mode squeezed vacuum [J]. Phys. Rev. A, 2002, 65: 062306.
[43] T. Opatrny, G. Kurizki and D. G. Welsch. Improvement on teleportation of continuous variables by photon subtraction via conditional measurement [J]. Phys. Rev. A, 2000, 61: 032302.
[44] A. Kitagawa, M. Takeoka, K. Wakui and M. Sasaki. Quantum gloves: Quantum states that encode as much as possible chirality and nothing else [J]. Phys. Rev. A, 2005, 72: 022304.
[45] D. E. Browne, J. Eisert, S. Scheel and M. B. Plenio. Driving non-Gaussian to Gaussian states with linear optics [J]. Phys. Rev. A, 2003, 67: 062320.
[46] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen. Algebraic approch to master equations with superoperator generators of su.1,1. and su.2. Lie algebras [J]. Phys. Rev. A, 2003, 67: 024101.
[47] S. J. Wu and Y. D. Zhang. Calculating the relative entropy of entangle- ment [J]. Phys. Rev. A, 2004, 61: 012308.
[48] N. Lutkenhaus, J. Calsamiglia, and K.-A. Suominen. Bell measurements for teleportation [J]. Phys. Rev. A, 1999, 59: 3295.
[49] L. Vaidman, and N. Yoran. Methods for reliable teleportation [J]. Phys. Rev. A, 1999, 59: 116.
[50] N. Korolkova, et.al.. Polarization squeezing and continuous-variable polarization entanglement [J]. Phys. Rev. A, 2002, 65: 052306.
[51] C. W. Gardiner and P. Zoller Quantum Noise, [M], 2nd edn. Berlin: Springer,2000.