黑洞时空中的量子纠缠提纯
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摘要
量子信息科学从诞生到现在虽只有10多年时间,但由于其重大的理论意义及广阔的应用前景而得以迅猛发展,扩展形成一片新兴的领域,并取得了引人瞩目的成就。量子纠缠是量子信息理论中的一个重要概念,它不但具有十分重要的理论意义,而且可以作为主要的资源应用于很多重要的量子信息操作中,如量子计算、量子通信等。广义相对论是物质世界的时空理论和时空中物质运动的普遍理论。黑洞物理是广义相对论理论中的一个重要部分,它联系了引力理论所描述的宏观世界和量子力学描述的微观世界。黑洞某些性质揭示了量子论和广义相对论之间的内在联系,在对量子信息理论的研究方面也具有十分重要的意义。近年来,在广义相对论框架下研究量子信息逐渐引起越来越多的人的兴趣,因为它不仅能进一步加深我们对量子信息的理解,而且还会在黑洞熵和黑洞信息疑难的研究中扮演重要的角色。作为广义相对论和量子信息的交叉学科,弯曲时空中量子信息已成为量子信息研究的新热点领域。我们将在广义相对论描述的弯曲时空中进行量子纠缠提纯的研究。
本文讨论了史瓦西时空背景下狄拉克场的纠缠提纯问题。分析了霍金温度、场的能量对纠缠提纯的影响。研究发现对于初始的两粒子Werner态,在平直时空中,只有其态参数,F(0 本文还探讨了当初始的Werner态由一般纠缠态组成时,其态参数α对纠缠提纯的影响。我们发现,参数α的取值对初始态的纠缠度大小有很明显的影响,并且在弯曲时空中会影响纠缠的衰减趋势,但在黑洞背景下,初始态有纠缠的范围并不受α的影响。
Recently ten years, quantum information has developed very quickly because of it's major theoretical significance and broad application prospect. Many emerg-ing fields have produced and a greatly achievement has been obtained. Quantum entanglement is both very important in the study of quantum infomation and is a major resource in quantum information, such as quantum teleportation and quan-tum computation and so on. On the other hand, general relativity is a theory about; spacetime and the movement of material. Black hole physics is an impor-tant part of the general relativity, which relates the theory of macroscopic world and microscopic world. So black hole physics have an important significance in the research of quantum imformation. Recently, more and more people are interested in reseach of the entanglement in a curved spaectime because it; is not only help us understand quantum information more clearly, but also very important in the black hole information paradox and the black hole entropy. Thus, considetable effort has been expended on the study of quantum entanglement in the curved spacetime. This thesis is devoted to the research of entanglement distillability in the black-hole spacetime
The entanglement distillability for the Dirac field in the spacetime of the Schwarzschild black hole is investigated. For the prepared Werner state in the innertial frame, it is entangled for 1/2< F< 1. But in the spacetime of black hole, the state is entangled for T< F< 1 (where r=(?)). This is unlike the case of innertial frame. We found that the Hawking temperature T and the energyωof the state play an important role in the low boundary T. T increases with the rise of the temperature T but it reduces with the rise of theω. Whenω→0 or T→∞, we have T→0.57.
We also investigate how the parameterαinfluences the entanglement dis-tillability when the prepared werner state is composed with uncertain entangled states. We found that although the parameterαaffects the entanglement, it does not change the range of the parameter F in which the state is entangled.
本文讨论了史瓦西时空背景下狄拉克场的纠缠提纯问题。分析了霍金温度、场的能量对纠缠提纯的影响。研究发现对于初始的两粒子Werner态,在平直时空中,只有其态参数,F(0
Recently ten years, quantum information has developed very quickly because of it's major theoretical significance and broad application prospect. Many emerg-ing fields have produced and a greatly achievement has been obtained. Quantum entanglement is both very important in the study of quantum infomation and is a major resource in quantum information, such as quantum teleportation and quan-tum computation and so on. On the other hand, general relativity is a theory about; spacetime and the movement of material. Black hole physics is an impor-tant part of the general relativity, which relates the theory of macroscopic world and microscopic world. So black hole physics have an important significance in the research of quantum imformation. Recently, more and more people are interested in reseach of the entanglement in a curved spaectime because it; is not only help us understand quantum information more clearly, but also very important in the black hole information paradox and the black hole entropy. Thus, considetable effort has been expended on the study of quantum entanglement in the curved spacetime. This thesis is devoted to the research of entanglement distillability in the black-hole spacetime
The entanglement distillability for the Dirac field in the spacetime of the Schwarzschild black hole is investigated. For the prepared Werner state in the innertial frame, it is entangled for 1/2< F< 1. But in the spacetime of black hole, the state is entangled for T< F< 1 (where r=(?)). This is unlike the case of innertial frame. We found that the Hawking temperature T and the energyωof the state play an important role in the low boundary T. T increases with the rise of the temperature T but it reduces with the rise of theω. Whenω→0 or T→∞, we have T→0.57.
We also investigate how the parameterαinfluences the entanglement dis-tillability when the prepared werner state is composed with uncertain entangled states. We found that although the parameterαaffects the entanglement, it does not change the range of the parameter F in which the state is entangled.
引文
[1]D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation[M]. (Springer-Verlag, Berlin),2000.
[2]张永德,量子信息物理原理[M].北京：科学出版社,2005.
[3]S. Chandrasekhar, The Mathematical Theory of Black Holes[M]. New York: Oxford University Press,1983.
[4]王永久,黑洞物理学[M].长沙：湖南师范大学出版社,2000.
[5]A. Peres and D. R. Terno, Quantum information and relativity theory[J]. Rev. Mod. Phys.2004,76:93.
[6]I. Fuentes-Schuller and R. B. Mann, Alice Falls into a Black Hole:Entangle-ment in Noninertial Frames [J]. Phys. Rev. Lett.2005.95:120404.
[7]P. M. Alsing. I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier. Entanglement of Dirac fields in noninertial frames[J]. Phys. Rev. A 2006.74:032326.
[8]P. M. Alsing and G. J. Milburn, Teleportation with a Uniformly Accelerated Partner[J]. Phys. Rev. Lett.2003,91:180404. [9] P. M. Alsing, D. McMahon. and G. J. Milburn. Teleportation in a non-inertial frame [J]. J. Opt. B:Quantum Semiclass. Opt.2004.6:S834.
[10]David E. Bruschi, Jorma Lonko, Eduardo Mart i n-Mart i nez, Andrzej Dragan, lvette Fuentes. Unruh effect in quantum information beyond the single-mode approximation[J]. Phys. Rev. A 2010.82:042332.
[11]P. C. W. Davies. Scalar production in Schwarzschild and Rindler metrics[J]. J. Phys. A 1975.8:609.
[12]W. G. Unruh. Notes on black-hole evaporation[J]. Phys. Rov. D 1976.11:870.
[13]Qiyuan Pan and Jiliang Jing. Degradation of nonmaximal entanglement of scalar and Dirac fields in noninertial frames[J]. Phys. Rev. A 2008.77: 024302.
[14]L. Lamata. M. A. Martin-Delgado and E. Solano, Relativity and Lorentz In-variance of Entanglement Distillability[J]. Phys. Rev. Lett.2006,97:250502.
[15]D. Ahn and M. S. Kim, Hawking-Unruh effect and the entanglement of two-mode squeezed states in Riemannian spacetime[J]. Phys. Lett. A 2007, 366:202.
[16]Yi Ling, Song He, Weigang Qiu, and Hongbao Zhang, Quantum entanglement of electromagnetic field in non-inertial reference frames[J]. J. Phys. A: Math. Theor.2007,40:9025.
[17]G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Continuous-variable entangle-ment sharing in noninertial frames [J]. Phys. Rev. A 2007,76:062112.
[18]Juan Leon and Eduardo Martin-Martinez, Spin and occupation number en-tanglement of Dirac fields for noninertial observers[J]. Phys. Rev. A 2009, 80:012314.
[19]R. B. Mann and V. M. Villalba. Speeding up entanglement degradation[J]. Phys. Rev. A 2009,80:022305.
[20]Andre G..S. Landulfo and George E. A. Matsas, Sudden death of entanglement and teleportation fidelity loss via the Unruh effect [J]. Phys. Rev. A 2009, 80:032315.
[21]J. Donkas and L. C. L. Hollenberg, Loss of spin entanglement for accelerated electrons in electric and magnetic fields[J]. Phys. Rev. A 2009,79:052109.
[22]S. Moradi, Distillability of entanglement in accelerated frames[J]. Phys. Rev. A 2009.79:064301.
[23]S. W. Hawking, Particle creation by black holes[J]. Comrnun Math. Phys. 1975.43:199.
[24]S. W. Hawking, Breakdown of predictability in gravitational collapse[J]. Phys. Rev. D 1976,14:2460.
[25]H. Terashima. Entanglement entropy of the black hole horizon[J]. Phys. Rev. D 2000,61:104016.
[26]R. Ruffini and.J. A. Wheeler. Introducing the black hole [J]. Phys. Today.1.971, 24(1):30.
[27]Xian-Hui Ge and Sang Pyo Kim. Quantum Entanglement and Teleporta-tion in Higher Dimensional Black Hole Spacetimes[J]. arXiv,2007. quant-ph/0707.4523.
[28]Xian-Hui Ge and You-Gen Shen, Teleportation in the Background of Schwarzschild Space-time[J]. Phys. Lett. B 2005,606:184.
[29]Qiyuan Pan and Jiliang Jing, Hawking radiation, Entanglement and Tele-portation in background of an asymptotically flat static black hole[J]. Phys. Rev. D 2008,78:065015.
[30]T. C. Ralph, G. J. Milburn, and T. Downes, Quantum connectivity of space-time and gravitationally induced decorrelation of entanglement [J]. Phys. Rev. A 2009,79:022121.
[31]E. Mart i n-Mart i nez. L.J. Garay, Juan Le 6 n, Unveiling quantum en-tanglement degradation near a Schwarzschild black hole[J]. Phys. Rev. D 2010.82:064006
[32]L. Bombelli. R. K. Koul, J. Lee, and R. Sorkin. Quantum source of entropy for black holes[J]. Phys. Rev. D 1986.34:373.
[33]C. Callan and F. Wilczek, On geometric entropy[J]. Phys. Lett, B 1994,333: 55.
[34]O. Dreyer. Quasinormal Modes, the Area Spectrum, and Black Hole En-tropy[J]. Phy. Rev. Lett.2003.90:081301.
[35]Jieei Wang. Qiyuan Pan. Songbai Chen and Jiliang Jing. Entanglement of cou-pled massive scalar field in the background of dilaton black hole[J]. Phys. Lett. B 2009,186:677.
[36]刘辽,赵峥, 广义相对论[M].高等教育出版社2004.
[37]H. T. Cho, Dirac quasinormal modes in Schwarzschild black hole space-times[J]. Phys. Rev. D 2003, 68:024003.
[38]Jiliang.Jing. Dirac quasinormal modes of Schwarzschild black hole[J]. Phys. Rev. D 2005,71:124006.
[39]Jiliang Jing, Late-time behavior of massive Dirac fields in a Schwarzschild background[J] Phys. Rev. D 2004.70:065004.
[40]M. Han, J. S. Olson, and J. P. Dowling, Generating entangled photons from the vacuum by accelerated measurements:Quantum-information theory and the Unruh-Davies effect [J]. Phys. Rev. A 2008,78:022302.
[41]David C. M. Ostapchuk and Robert B. Mann, Generating entangled fermions by accelerated measurements on the vacuum[J]. Phys. Rev. A 2009.79: 042333.
[42]M. D. Kruskal, Maximal extension of Schwarzschild metric[J]. Phys. Rev. 1960,119:1743.
[43]N. N. Bogoliubov, New method in the theory of superconductivity. Ⅱ[J]. Sov. Phys. JETP 1958,7:51.
[44]A. Einstein, B. Podolsky. and N. Rosen, Can Quantum-Mechanical Descrip-tion of Physical Reality Be Considered Complete?[J]. Phys. Rev.1935,47: 777.
[45]D. Bosehi, S. Branca, F. De Martini. L. Hardy, and S. Popescn, Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels[J]. Phys. Rev. Lett.1998, 80:1121.
[46]J. W. Pan, C. Simon, C. Brukner. and A. Zeilinger. Entanglement purification for quantum communication[J]. Nature(London) 2001,410:1067.
[47]E. Schrodinger. Die Gegenwartige Situation in der Quanten-Mechanik[J]. Naturwissenschaften 1935,23:807.
[48]A. Peres, Separability Criterion for Density Matrices[J]. Phys. Rev. Lett. 1996,77:1413.
[49]M. Horodecki, P. Horodecki. and R. Horodecki, Separability of mixed states: necessary and sufficient conditions[J]. Phys. Lett. A 1996,223:1.
[50]Shengjun Wu, Xuemei Chen, and Yongde Zhang. A necessary and sufficient criterion for multipartite separable states[J]. Phys. Lett. A 2000,275:244.
[51]B. M. Terhal, Bell inequalities and the separability criterion[J]. Phys. Lett. A 2000,271:319.
[52]M. A. Nielsen and I. Chuang, Quantum Computation and Quantum infor-mation[M]. Cambridge:Cambridge University Press,2000.
[53]C. H. Bennett, H. J. Bernstein, S. Popeseu, and B. Schumacher. Concentrating partial entanglement by local operations[J]. Phys. Rev. A 1996.53:2046.
[54]G. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels[J]. Phys. Rev. Lett.1996,76:722.
[55]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.
[56]G. Vidal and R. F. Werner, Computable measure of entanglement [J]. Phys. Rev. A 2002.65:032314.
[57]M. B. Plenio, Logarithmic Negativity:A Full Entanglement Monotone That is not Convex[J]. Phys. Rev. Lett.2005.95:090503.
[58]E. M. Rains, Rigorous treatment of distillable entanglement [J]. Phys. Rev. A 1999.60:173.
[59]W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits[J]. Phys. Rev. Lett 1998,80:2245.
[60]V. Coffman, J. Kundu, and W.K. Wootters. Distributed entanglement [J]. Phys. Rev. A 2000.61:052306.
[61]R. S. Ingarden, A. Kossakowski, and M. Ohya, Information Dynamics and Open Systems-Classical and Quantum Approach[M]. Dordrecht:Kluwer Academic Publishers,1997.
[62]D.R. Brill and J.A. Wheeler, Interaction of Neutrinos and Gravitational Fields[J]. Rev. Mod. Phys.1957,29:465; 1992,45:3888(E).
[63]T. Damoar and R. R.uffini, Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism[J]. Phys. Rev. D 1976,14:332.
[64]Jieci Wang, Qiyuan Pang, Jiliang Jing, Projective measurements and gen-eration of entangled Dirac particles in Schwarzschild Spacetime [J]. Ann. Phys.2010,325:1190
[65]N. D. Birrell and P. C. W. Davies, Quantum fields in curved space[M]. New York:Cambridge University Press,1982.
[66]S. Sannan, Heuristic derivation of the probability distributions of particles emitted by a black hole[J]. Gen. Rel. Grav.1988,20:239.
[67]Z. Zhao and Y. X. Gui. The connection between Unruh scheme and Damour-Ruffini scheme in rindler space-time and η-ε space-time[J]. IL Nuovo Cimento B 1994,109:355.
[68]S. M. Barniett and P. M. Radmore, Methods in Theoretical Quantum Optics, 67-80[M]. (Oxford University Press, New York),1997.
[69]D. Ahn, Y. H. Moon. R. B. Mann and 1. Fuentes-Schuller. The black hole final state for the Dirac fields In Schwarzschild spacetime [J]. JHEP 2008,0806: 062.
[70]R. Kerner and R. B. Mann. Tunnelling, temperature, and Taub-NUT black holes[J]. Phys. Rev. D 2006,73:104010.
[71]Qing-Quan Jiang. Shuang-Qing Wu. and Xu Cai, Hawking radiation from dila-tonic black holes via anomalies[J]. Phys. Rev. D 2007.75:064029.
[72]R.F.Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model[J]. Phys. Rev. A 1989.40:4277.
[2]张永德,量子信息物理原理[M].北京：科学出版社,2005.
[3]S. Chandrasekhar, The Mathematical Theory of Black Holes[M]. New York: Oxford University Press,1983.
[4]王永久,黑洞物理学[M].长沙：湖南师范大学出版社,2000.
[5]A. Peres and D. R. Terno, Quantum information and relativity theory[J]. Rev. Mod. Phys.2004,76:93.
[6]I. Fuentes-Schuller and R. B. Mann, Alice Falls into a Black Hole:Entangle-ment in Noninertial Frames [J]. Phys. Rev. Lett.2005.95:120404.
[7]P. M. Alsing. I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier. Entanglement of Dirac fields in noninertial frames[J]. Phys. Rev. A 2006.74:032326.
[8]P. M. Alsing and G. J. Milburn, Teleportation with a Uniformly Accelerated Partner[J]. Phys. Rev. Lett.2003,91:180404. [9] P. M. Alsing, D. McMahon. and G. J. Milburn. Teleportation in a non-inertial frame [J]. J. Opt. B:Quantum Semiclass. Opt.2004.6:S834.
[10]David E. Bruschi, Jorma Lonko, Eduardo Mart i n-Mart i nez, Andrzej Dragan, lvette Fuentes. Unruh effect in quantum information beyond the single-mode approximation[J]. Phys. Rev. A 2010.82:042332.
[11]P. C. W. Davies. Scalar production in Schwarzschild and Rindler metrics[J]. J. Phys. A 1975.8:609.
[12]W. G. Unruh. Notes on black-hole evaporation[J]. Phys. Rov. D 1976.11:870.
[13]Qiyuan Pan and Jiliang Jing. Degradation of nonmaximal entanglement of scalar and Dirac fields in noninertial frames[J]. Phys. Rev. A 2008.77: 024302.
[14]L. Lamata. M. A. Martin-Delgado and E. Solano, Relativity and Lorentz In-variance of Entanglement Distillability[J]. Phys. Rev. Lett.2006,97:250502.
[15]D. Ahn and M. S. Kim, Hawking-Unruh effect and the entanglement of two-mode squeezed states in Riemannian spacetime[J]. Phys. Lett. A 2007, 366:202.
[16]Yi Ling, Song He, Weigang Qiu, and Hongbao Zhang, Quantum entanglement of electromagnetic field in non-inertial reference frames[J]. J. Phys. A: Math. Theor.2007,40:9025.
[17]G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Continuous-variable entangle-ment sharing in noninertial frames [J]. Phys. Rev. A 2007,76:062112.
[18]Juan Leon and Eduardo Martin-Martinez, Spin and occupation number en-tanglement of Dirac fields for noninertial observers[J]. Phys. Rev. A 2009, 80:012314.
[19]R. B. Mann and V. M. Villalba. Speeding up entanglement degradation[J]. Phys. Rev. A 2009,80:022305.
[20]Andre G..S. Landulfo and George E. A. Matsas, Sudden death of entanglement and teleportation fidelity loss via the Unruh effect [J]. Phys. Rev. A 2009, 80:032315.
[21]J. Donkas and L. C. L. Hollenberg, Loss of spin entanglement for accelerated electrons in electric and magnetic fields[J]. Phys. Rev. A 2009,79:052109.
[22]S. Moradi, Distillability of entanglement in accelerated frames[J]. Phys. Rev. A 2009.79:064301.
[23]S. W. Hawking, Particle creation by black holes[J]. Comrnun Math. Phys. 1975.43:199.
[24]S. W. Hawking, Breakdown of predictability in gravitational collapse[J]. Phys. Rev. D 1976,14:2460.
[25]H. Terashima. Entanglement entropy of the black hole horizon[J]. Phys. Rev. D 2000,61:104016.
[26]R. Ruffini and.J. A. Wheeler. Introducing the black hole [J]. Phys. Today.1.971, 24(1):30.
[27]Xian-Hui Ge and Sang Pyo Kim. Quantum Entanglement and Teleporta-tion in Higher Dimensional Black Hole Spacetimes[J]. arXiv,2007. quant-ph/0707.4523.
[28]Xian-Hui Ge and You-Gen Shen, Teleportation in the Background of Schwarzschild Space-time[J]. Phys. Lett. B 2005,606:184.
[29]Qiyuan Pan and Jiliang Jing, Hawking radiation, Entanglement and Tele-portation in background of an asymptotically flat static black hole[J]. Phys. Rev. D 2008,78:065015.
[30]T. C. Ralph, G. J. Milburn, and T. Downes, Quantum connectivity of space-time and gravitationally induced decorrelation of entanglement [J]. Phys. Rev. A 2009,79:022121.
[31]E. Mart i n-Mart i nez. L.J. Garay, Juan Le 6 n, Unveiling quantum en-tanglement degradation near a Schwarzschild black hole[J]. Phys. Rev. D 2010.82:064006
[32]L. Bombelli. R. K. Koul, J. Lee, and R. Sorkin. Quantum source of entropy for black holes[J]. Phys. Rev. D 1986.34:373.
[33]C. Callan and F. Wilczek, On geometric entropy[J]. Phys. Lett, B 1994,333: 55.
[34]O. Dreyer. Quasinormal Modes, the Area Spectrum, and Black Hole En-tropy[J]. Phy. Rev. Lett.2003.90:081301.
[35]Jieei Wang. Qiyuan Pan. Songbai Chen and Jiliang Jing. Entanglement of cou-pled massive scalar field in the background of dilaton black hole[J]. Phys. Lett. B 2009,186:677.
[36]刘辽,赵峥, 广义相对论[M].高等教育出版社2004.
[37]H. T. Cho, Dirac quasinormal modes in Schwarzschild black hole space-times[J]. Phys. Rev. D 2003, 68:024003.
[38]Jiliang.Jing. Dirac quasinormal modes of Schwarzschild black hole[J]. Phys. Rev. D 2005,71:124006.
[39]Jiliang Jing, Late-time behavior of massive Dirac fields in a Schwarzschild background[J] Phys. Rev. D 2004.70:065004.
[40]M. Han, J. S. Olson, and J. P. Dowling, Generating entangled photons from the vacuum by accelerated measurements:Quantum-information theory and the Unruh-Davies effect [J]. Phys. Rev. A 2008,78:022302.
[41]David C. M. Ostapchuk and Robert B. Mann, Generating entangled fermions by accelerated measurements on the vacuum[J]. Phys. Rev. A 2009.79: 042333.
[42]M. D. Kruskal, Maximal extension of Schwarzschild metric[J]. Phys. Rev. 1960,119:1743.
[43]N. N. Bogoliubov, New method in the theory of superconductivity. Ⅱ[J]. Sov. Phys. JETP 1958,7:51.
[44]A. Einstein, B. Podolsky. and N. Rosen, Can Quantum-Mechanical Descrip-tion of Physical Reality Be Considered Complete?[J]. Phys. Rev.1935,47: 777.
[45]D. Bosehi, S. Branca, F. De Martini. L. Hardy, and S. Popescn, Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels[J]. Phys. Rev. Lett.1998, 80:1121.
[46]J. W. Pan, C. Simon, C. Brukner. and A. Zeilinger. Entanglement purification for quantum communication[J]. Nature(London) 2001,410:1067.
[47]E. Schrodinger. Die Gegenwartige Situation in der Quanten-Mechanik[J]. Naturwissenschaften 1935,23:807.
[48]A. Peres, Separability Criterion for Density Matrices[J]. Phys. Rev. Lett. 1996,77:1413.
[49]M. Horodecki, P. Horodecki. and R. Horodecki, Separability of mixed states: necessary and sufficient conditions[J]. Phys. Lett. A 1996,223:1.
[50]Shengjun Wu, Xuemei Chen, and Yongde Zhang. A necessary and sufficient criterion for multipartite separable states[J]. Phys. Lett. A 2000,275:244.
[51]B. M. Terhal, Bell inequalities and the separability criterion[J]. Phys. Lett. A 2000,271:319.
[52]M. A. Nielsen and I. Chuang, Quantum Computation and Quantum infor-mation[M]. Cambridge:Cambridge University Press,2000.
[53]C. H. Bennett, H. J. Bernstein, S. Popeseu, and B. Schumacher. Concentrating partial entanglement by local operations[J]. Phys. Rev. A 1996.53:2046.
[54]G. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels[J]. Phys. Rev. Lett.1996,76:722.
[55]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.
[56]G. Vidal and R. F. Werner, Computable measure of entanglement [J]. Phys. Rev. A 2002.65:032314.
[57]M. B. Plenio, Logarithmic Negativity:A Full Entanglement Monotone That is not Convex[J]. Phys. Rev. Lett.2005.95:090503.
[58]E. M. Rains, Rigorous treatment of distillable entanglement [J]. Phys. Rev. A 1999.60:173.
[59]W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits[J]. Phys. Rev. Lett 1998,80:2245.
[60]V. Coffman, J. Kundu, and W.K. Wootters. Distributed entanglement [J]. Phys. Rev. A 2000.61:052306.
[61]R. S. Ingarden, A. Kossakowski, and M. Ohya, Information Dynamics and Open Systems-Classical and Quantum Approach[M]. Dordrecht:Kluwer Academic Publishers,1997.
[62]D.R. Brill and J.A. Wheeler, Interaction of Neutrinos and Gravitational Fields[J]. Rev. Mod. Phys.1957,29:465; 1992,45:3888(E).
[63]T. Damoar and R. R.uffini, Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism[J]. Phys. Rev. D 1976,14:332.
[64]Jieci Wang, Qiyuan Pang, Jiliang Jing, Projective measurements and gen-eration of entangled Dirac particles in Schwarzschild Spacetime [J]. Ann. Phys.2010,325:1190
[65]N. D. Birrell and P. C. W. Davies, Quantum fields in curved space[M]. New York:Cambridge University Press,1982.
[66]S. Sannan, Heuristic derivation of the probability distributions of particles emitted by a black hole[J]. Gen. Rel. Grav.1988,20:239.
[67]Z. Zhao and Y. X. Gui. The connection between Unruh scheme and Damour-Ruffini scheme in rindler space-time and η-ε space-time[J]. IL Nuovo Cimento B 1994,109:355.
[68]S. M. Barniett and P. M. Radmore, Methods in Theoretical Quantum Optics, 67-80[M]. (Oxford University Press, New York),1997.
[69]D. Ahn, Y. H. Moon. R. B. Mann and 1. Fuentes-Schuller. The black hole final state for the Dirac fields In Schwarzschild spacetime [J]. JHEP 2008,0806: 062.
[70]R. Kerner and R. B. Mann. Tunnelling, temperature, and Taub-NUT black holes[J]. Phys. Rev. D 2006,73:104010.
[71]Qing-Quan Jiang. Shuang-Qing Wu. and Xu Cai, Hawking radiation from dila-tonic black holes via anomalies[J]. Phys. Rev. D 2007.75:064029.
[72]R.F.Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model[J]. Phys. Rev. A 1989.40:4277.