自由空间量子密钥分配协议研究
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摘要
量子密码学是基于量子力学中的海森堡测不准原理和量子不可克隆原理发展起来的一种新型保密通信技术,在理论上已被证明是绝对安全、不可破译的。由于其绝对安全性,在军事、外交、通信、电子商务等领域具有广泛的应用前景。
近年来的研究表明:光束除了具有自旋角动量(spin angular momentum, SAM),还具有由螺旋型相位结构产生的轨道角动量(orbital angular momentum, OAM),且与极化量子态(自旋角动量)不同,OAM态可同时包含任意多种状态。
本文在BB84量子密钥分配协议(quantum key distribution, QKD)基础上,将OAM应用到QKD过程中,获得一个多维空间QKD方案,并对其进行数值仿真。仿真结果表明,基于OAM的QKD协议极大地提高了密钥生成效率,并且不需要调整参考系,适合自由空间通信;随后,本文将互无偏基概念与OAM纠缠概念相结合,获得两种适合自由空间通信的QKD协议,分别是基于纠缠特性的随机相位QKD协议和基于OAM态纠缠的QKD协议,给出其具体实验实现过程,通过数值仿真和实验验证了协议的可行性;由于OAM是空间波函数,当OAM作为量子信息载体,在自由空间传输时不可避免受到大气湍流的影响,因此,论文最后研究了大气湍流随机信道的统计特性,获得大气湍流的数学模型,以Labview为数值计算工具,分析大气湍流对OAM状态的影响,获得所提出协议的量子误码率性能。论文对自由空间地面点对点大气光链路中QKD实验,以及卫星-地面量子通信的研究具有一定的参考作用。
Quantum cryptography is a new type of secure communication technologies that based on Heisenberg uncertainty principle and no-cloning principle of quantum mechanics. It has been proved to be absolutely safe in theory. Because of its absolute security, quantum cryptography has a wide range of applications in military, diplomatic, communications, e-commerce and other fields.
It is shown that the light has both spin angular momentum (SAM) and orbital angular momentum (OAM), different from the SAM, the OAM state can contain any number of states.
Based on BB84 quantum key distribution (QKD) protocol, we first apply OAM to QKD scheme and obtain a new multi-dimensional QKD protocol. The simulation results show that the protocol greatly improves the efficiency of key generation, and the states used in the transmission are invariant under rotations about the propagation direction, independent of the alignment between the reference frames of the sender and receiver. Hence the protocol appeals for free space communication. Secondly, we obtain two free-space QKD protocols by combining the concept of mutual unbiased basis and the entanglement. They are named random phase modulation QKD protocol and entanglement-based on OAM state QKD protocol. We describe their implementation process in detail in the article, and the two protocols’feasibilities are verified by the simulation and experimental. At last, since OAM states are the spatial wave function, they are inevitably impacted by atmospheric turbulence, so we present the simulation of atmospheric aberration, and analyze the impact of the atmospheric turbulence on the proposed QKD protocols. The article can be a reference for the study of quantum communications in free space and in interstellar communication.
近年来的研究表明:光束除了具有自旋角动量(spin angular momentum, SAM),还具有由螺旋型相位结构产生的轨道角动量(orbital angular momentum, OAM),且与极化量子态(自旋角动量)不同,OAM态可同时包含任意多种状态。
本文在BB84量子密钥分配协议(quantum key distribution, QKD)基础上,将OAM应用到QKD过程中,获得一个多维空间QKD方案,并对其进行数值仿真。仿真结果表明,基于OAM的QKD协议极大地提高了密钥生成效率,并且不需要调整参考系,适合自由空间通信;随后,本文将互无偏基概念与OAM纠缠概念相结合,获得两种适合自由空间通信的QKD协议,分别是基于纠缠特性的随机相位QKD协议和基于OAM态纠缠的QKD协议,给出其具体实验实现过程,通过数值仿真和实验验证了协议的可行性;由于OAM是空间波函数,当OAM作为量子信息载体,在自由空间传输时不可避免受到大气湍流的影响,因此,论文最后研究了大气湍流随机信道的统计特性,获得大气湍流的数学模型,以Labview为数值计算工具,分析大气湍流对OAM状态的影响,获得所提出协议的量子误码率性能。论文对自由空间地面点对点大气光链路中QKD实验,以及卫星-地面量子通信的研究具有一定的参考作用。
Quantum cryptography is a new type of secure communication technologies that based on Heisenberg uncertainty principle and no-cloning principle of quantum mechanics. It has been proved to be absolutely safe in theory. Because of its absolute security, quantum cryptography has a wide range of applications in military, diplomatic, communications, e-commerce and other fields.
It is shown that the light has both spin angular momentum (SAM) and orbital angular momentum (OAM), different from the SAM, the OAM state can contain any number of states.
Based on BB84 quantum key distribution (QKD) protocol, we first apply OAM to QKD scheme and obtain a new multi-dimensional QKD protocol. The simulation results show that the protocol greatly improves the efficiency of key generation, and the states used in the transmission are invariant under rotations about the propagation direction, independent of the alignment between the reference frames of the sender and receiver. Hence the protocol appeals for free space communication. Secondly, we obtain two free-space QKD protocols by combining the concept of mutual unbiased basis and the entanglement. They are named random phase modulation QKD protocol and entanglement-based on OAM state QKD protocol. We describe their implementation process in detail in the article, and the two protocols’feasibilities are verified by the simulation and experimental. At last, since OAM states are the spatial wave function, they are inevitably impacted by atmospheric turbulence, so we present the simulation of atmospheric aberration, and analyze the impact of the atmospheric turbulence on the proposed QKD protocols. The article can be a reference for the study of quantum communications in free space and in interstellar communication.
引文
[1]冯登国.国内外密码学研究现状及发展趋势[C].通信学报, 2002, 23(5): 18-26
[2]张镇九,张昭理,李爱民.量子计算与通信加密[C].华中师范大学出版社,2002, 259-263
[3] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer [J]. SIAM J. Comput., 1999, 41:303-332
[4] D. Busrs, N. Lkutehnuas. Quanutm key distribution: from principles to practicalies [J]. AAECC, 2000, 10: 383-399
[5] N. Gisin, G. Wolgfgang, et.al. Quantum cryptography [J]. Phys. Rev. Lett., 2002, 74: 145-195
[6] D. Gottesman, Hoi-Kwong Lo. Proof of security of quantum key distribulion with two-way classical communications [J]. IEEE Transactions on Information Theory, 2003, 49: 457-475
[7] Hoi-Kwong Lo, H. F. Chau. Unconditional security of quantum key distribution over arbitrarily long distances [J]. Science, 1999, 283: 2050-2056
[8]张启仁.量子力学[M].高等教育出版社, 1989
[9] C. H. Bennett and G. Brassard. Quantum cryptography:’public key distribution and coin tossing’[C]. Presented at the IEEE Conf. Computers, Systems Signal Processing, Bangalore, India, Dec. 10-12, 1984
[10] C. H. Bennett. Quantum cryptography using any two nonorthogonal states [J]. Phys. Rev. Lett., 1992, 68: 3121-3124
[11] A. K. Ekert. Quantum cryptography based on Bell’s theorem [J]. Phys. Rev. Lett., 1991, 67: 661-663
[12] D. Patrick. Kumavor, C. Alan. Comparison of four multi-user quantum key distribution schemes over passive optical networks [J]. Journal of Lightwave Technology, January 2005, 23: 268-276
[13] P. Townsend. Quantum cryptography on multiuser optical fiber networks [J]. Nature, 1997, 385: 47-49
[14] P. Villoresi T, Jennewein F, Tamburini et al. Experimental verification of the feasibility of a quantum channel between space and earth [J]. Phys. Rev. Lett., 2008, 10: 033-038
[15] B. C. Jacobs, J. D. Franson. Quantum cryptography in free space [J]. Optics Letters, 1996,21: 1854-1856
[16] W. T. Buttler, R. J. Hughes, P. G.. Kwiat, et al. Free-space quantum key distribution [J]. Phys. Rev. Lett., 1998, 57: 2379-2382
[17] R. J. Hughes, W. T. Buttler, P. G.. Kwiat, et al. Quantum cryptography for secure free-space communications [J]. Proceedings of SPIE, 1999, 3615: 98-103
[18] R. J. Hughes, W. T. Buttler, P. G.. Kwiat, et al. Free-space quantum cryptography in daylight [J]. Proceedings of SPIE, 2000, 3932: 117-126
[19] M. Furst, H. Weier, T. Schmitt-Manderbach, et al. Free-Space quantum key distribution over 144 km [J]. Proceedings of SPIE, 2006, 6399: 1-7
[20] L. Allen, M. J. Padgett, M. Babiker. The orbital angular momentum of light progress in Optics [J]. Science, 1999, 291-372
[21]吴健,杨春平,刘建斌.大气中的光传输理论[M].北京邮电大学出版社, 2005: 130-132
[22] V. S. L’vov. Universality of turbulence [J]. Nature, 1998, 396: 515-521
[23] A. N. Kolmogorov. A refinement of previous hypotheses concerning local structure of turbulence in a viscous incornpressible fluid at high Reynolds number [J]. Journal of Fluid Mechanics, 1962, 13: 82-85
[24] A. N. Kolmogorov. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers [J]. Science, 1941, 30: 301-305
[25] A. S. Monin, A. M. Obukhov. Basic laws of turbulent mixing in the atmosphere near the ground [C]. Tr Akad Nauk SSSR Geofiz Inst, 1954, 24: 163-187
[26] Theodore vov Karman. Progress in the statistical theory of turbulence [C]. Heat Transfer and Fluid Mechanics Institute, Los Angeles, California, June 23, 1948
[27] R. J. Hill. Theory of Saturation of optical scintillation by strong turbulence [J]. J. Opt. Soc. Am. A, 1981, 72: 212-222
[28] L. C. Andrews. Analytical model for the refranctive index power spectrum and its application to optical scintillations in the atmosphere [J]. Journal of Modern Optics, 1992, 39: 1849
[29]李承祖.量子通信和量子计算[M].北京国防科技大学出版社, 2000, 101-145, 255-349
[30] W. K. Wootters, W. H. Zurek. A single quantum cannot be cloned [J]. Nature 299, 1982, 802-803
[31] J. S. Bell. On the Einstein-Podolsky-Rosen paradox [J]. Phys. Rev. Lett., 1964, 1: 195-200
[32] J. F. Clauser, M. A. Horne, A. Shimony, et al. Proposed experiment to test local-hidden-variable theories [J]. Phys. Rev. Lett., 1969, 23: 880-884.
[33] C. H. Bennett. Quantum cryptography using any two nonorthogonal states [J]. Phys. Rev. Lett., 1992, 68: 3121-3124
[34] D. Bruss. Optimal eavesdropping in quantum cryptography with six states [J]. Phys. Rev. Lett. 1998, 81: 3018-3021
[35] D. Mayers. Unconditional security in quantum cryptography [J]. Journal of ACM, 2001, 48: 351-406
[36] P. W. Shor, J. Preskill. Simple proof of security of the BB84 quantum key distribution protocol [J]. Phys. Rev. Lett., 2000, 85: 441-444
[37] H. K. Lo. Proof of unconditional security of six-state quantum key distribution scheme [C]. Quantum Information and Computation (QIC), 2001, 1: 81-94
[38] Daniel Gottesman, Hoi-Kwong Lo. Proof of security of quantum key distribution with two-way classical communications [J]. IEEE Information theory society, 2003, 49: 457-475
[39] H. Inamori. Security of EPR-based Quantum Key Distribution using three bases [J]. Quantum physics, 2008, 1-12
[40] H. Bechmann-Pasquinucci and N. Gisin. Incoherent and coherent eavesdropping in the 6-state protocol of quantum [J]. Phys. Rev. Lett., 1999, 59: 38- 42
[41]苏志锟等.基于光子轨道角动量的密码通信方案研究[J].物理学报, May 2008, 57: 3016-3021
[42] G. A. Turubule, D. A. Robert, G. M. Smith. et al. The generation of free-space Laguerre-Gaussian modes at millimet re-wave frequencies by use of a spiral phaseplate [J]. Optics Communications, 1996, 127: 183-188
[43] J. Courtial, M. J. Padgett. Performance of a cylindrical lens made converter for producing Laguerre-Gascssian laser modes [J]. Optics Communications, 1999, 159: 13-18
[44] G. Gibson, J. Courtial, M. J. Padgett. Free-space information transfer using light beams carrying orbital angular momentum [J]. Optics Communications, 2004, 12: 5448-5456
[45] Zhi-Kun Su, Fa-Qiang Wang, et al. A simple scheme for quantum networks based on orbital angular momentum states of photons [J]. Optics Communications, 2008, 281: 5063-5066
[46] Mark T. Gruneisen, A. Warner. Miller, C. Raymond. Holographic generation of complexfields with spatial light modulators: application to quantum key distribution [J]. Optical society, May 2008, 4392(28): A32-A42
[47] M Fedetrico. Spedalieri. Quantum key distribution without reference frame alignment: Expoiting photon orbital angular momentum [J]. Optics Communications, 2006, 260: 340-346
[48] C. Archer. There is no generalization of known formulas for mutually unbiased bases [J]. Phys. Rev. Lett., 2005, 46: 022-106
[49] P. Sulc, J. Tolar. Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions [J]. Phys. Rev. Lett., 2007, 40: 15099-15111
[50] M. Bourennane et al. Quantum key distribution using multi level encoding: security analysis [J]. Phys. Rev. Lett., 2002, 10065(35)
[51] M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani. Experimental realization of any discrete unitary operator [J]. Phys. Rev. Lett., 1994, 73: 58-61
[52] I. Csisz, J. Korner. Broadcast channels with confidential messages [J]. IEEE Trans. Inf. Theory, 24, 1978, 339-348
[53] M. J. W. Hall. Information exclusion: principle for complement observables [J]. Phys. Rev. Lett., 1995, 74: 3307-3310
[54] D. Mayers. Unconditional security in quantum cryptography [J]. Comput. Mac, 2001, 48: 351-406
[2]张镇九,张昭理,李爱民.量子计算与通信加密[C].华中师范大学出版社,2002, 259-263
[3] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer [J]. SIAM J. Comput., 1999, 41:303-332
[4] D. Busrs, N. Lkutehnuas. Quanutm key distribution: from principles to practicalies [J]. AAECC, 2000, 10: 383-399
[5] N. Gisin, G. Wolgfgang, et.al. Quantum cryptography [J]. Phys. Rev. Lett., 2002, 74: 145-195
[6] D. Gottesman, Hoi-Kwong Lo. Proof of security of quantum key distribulion with two-way classical communications [J]. IEEE Transactions on Information Theory, 2003, 49: 457-475
[7] Hoi-Kwong Lo, H. F. Chau. Unconditional security of quantum key distribution over arbitrarily long distances [J]. Science, 1999, 283: 2050-2056
[8]张启仁.量子力学[M].高等教育出版社, 1989
[9] C. H. Bennett and G. Brassard. Quantum cryptography:’public key distribution and coin tossing’[C]. Presented at the IEEE Conf. Computers, Systems Signal Processing, Bangalore, India, Dec. 10-12, 1984
[10] C. H. Bennett. Quantum cryptography using any two nonorthogonal states [J]. Phys. Rev. Lett., 1992, 68: 3121-3124
[11] A. K. Ekert. Quantum cryptography based on Bell’s theorem [J]. Phys. Rev. Lett., 1991, 67: 661-663
[12] D. Patrick. Kumavor, C. Alan. Comparison of four multi-user quantum key distribution schemes over passive optical networks [J]. Journal of Lightwave Technology, January 2005, 23: 268-276
[13] P. Townsend. Quantum cryptography on multiuser optical fiber networks [J]. Nature, 1997, 385: 47-49
[14] P. Villoresi T, Jennewein F, Tamburini et al. Experimental verification of the feasibility of a quantum channel between space and earth [J]. Phys. Rev. Lett., 2008, 10: 033-038
[15] B. C. Jacobs, J. D. Franson. Quantum cryptography in free space [J]. Optics Letters, 1996,21: 1854-1856
[16] W. T. Buttler, R. J. Hughes, P. G.. Kwiat, et al. Free-space quantum key distribution [J]. Phys. Rev. Lett., 1998, 57: 2379-2382
[17] R. J. Hughes, W. T. Buttler, P. G.. Kwiat, et al. Quantum cryptography for secure free-space communications [J]. Proceedings of SPIE, 1999, 3615: 98-103
[18] R. J. Hughes, W. T. Buttler, P. G.. Kwiat, et al. Free-space quantum cryptography in daylight [J]. Proceedings of SPIE, 2000, 3932: 117-126
[19] M. Furst, H. Weier, T. Schmitt-Manderbach, et al. Free-Space quantum key distribution over 144 km [J]. Proceedings of SPIE, 2006, 6399: 1-7
[20] L. Allen, M. J. Padgett, M. Babiker. The orbital angular momentum of light progress in Optics [J]. Science, 1999, 291-372
[21]吴健,杨春平,刘建斌.大气中的光传输理论[M].北京邮电大学出版社, 2005: 130-132
[22] V. S. L’vov. Universality of turbulence [J]. Nature, 1998, 396: 515-521
[23] A. N. Kolmogorov. A refinement of previous hypotheses concerning local structure of turbulence in a viscous incornpressible fluid at high Reynolds number [J]. Journal of Fluid Mechanics, 1962, 13: 82-85
[24] A. N. Kolmogorov. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers [J]. Science, 1941, 30: 301-305
[25] A. S. Monin, A. M. Obukhov. Basic laws of turbulent mixing in the atmosphere near the ground [C]. Tr Akad Nauk SSSR Geofiz Inst, 1954, 24: 163-187
[26] Theodore vov Karman. Progress in the statistical theory of turbulence [C]. Heat Transfer and Fluid Mechanics Institute, Los Angeles, California, June 23, 1948
[27] R. J. Hill. Theory of Saturation of optical scintillation by strong turbulence [J]. J. Opt. Soc. Am. A, 1981, 72: 212-222
[28] L. C. Andrews. Analytical model for the refranctive index power spectrum and its application to optical scintillations in the atmosphere [J]. Journal of Modern Optics, 1992, 39: 1849
[29]李承祖.量子通信和量子计算[M].北京国防科技大学出版社, 2000, 101-145, 255-349
[30] W. K. Wootters, W. H. Zurek. A single quantum cannot be cloned [J]. Nature 299, 1982, 802-803
[31] J. S. Bell. On the Einstein-Podolsky-Rosen paradox [J]. Phys. Rev. Lett., 1964, 1: 195-200
[32] J. F. Clauser, M. A. Horne, A. Shimony, et al. Proposed experiment to test local-hidden-variable theories [J]. Phys. Rev. Lett., 1969, 23: 880-884.
[33] C. H. Bennett. Quantum cryptography using any two nonorthogonal states [J]. Phys. Rev. Lett., 1992, 68: 3121-3124
[34] D. Bruss. Optimal eavesdropping in quantum cryptography with six states [J]. Phys. Rev. Lett. 1998, 81: 3018-3021
[35] D. Mayers. Unconditional security in quantum cryptography [J]. Journal of ACM, 2001, 48: 351-406
[36] P. W. Shor, J. Preskill. Simple proof of security of the BB84 quantum key distribution protocol [J]. Phys. Rev. Lett., 2000, 85: 441-444
[37] H. K. Lo. Proof of unconditional security of six-state quantum key distribution scheme [C]. Quantum Information and Computation (QIC), 2001, 1: 81-94
[38] Daniel Gottesman, Hoi-Kwong Lo. Proof of security of quantum key distribution with two-way classical communications [J]. IEEE Information theory society, 2003, 49: 457-475
[39] H. Inamori. Security of EPR-based Quantum Key Distribution using three bases [J]. Quantum physics, 2008, 1-12
[40] H. Bechmann-Pasquinucci and N. Gisin. Incoherent and coherent eavesdropping in the 6-state protocol of quantum [J]. Phys. Rev. Lett., 1999, 59: 38- 42
[41]苏志锟等.基于光子轨道角动量的密码通信方案研究[J].物理学报, May 2008, 57: 3016-3021
[42] G. A. Turubule, D. A. Robert, G. M. Smith. et al. The generation of free-space Laguerre-Gaussian modes at millimet re-wave frequencies by use of a spiral phaseplate [J]. Optics Communications, 1996, 127: 183-188
[43] J. Courtial, M. J. Padgett. Performance of a cylindrical lens made converter for producing Laguerre-Gascssian laser modes [J]. Optics Communications, 1999, 159: 13-18
[44] G. Gibson, J. Courtial, M. J. Padgett. Free-space information transfer using light beams carrying orbital angular momentum [J]. Optics Communications, 2004, 12: 5448-5456
[45] Zhi-Kun Su, Fa-Qiang Wang, et al. A simple scheme for quantum networks based on orbital angular momentum states of photons [J]. Optics Communications, 2008, 281: 5063-5066
[46] Mark T. Gruneisen, A. Warner. Miller, C. Raymond. Holographic generation of complexfields with spatial light modulators: application to quantum key distribution [J]. Optical society, May 2008, 4392(28): A32-A42
[47] M Fedetrico. Spedalieri. Quantum key distribution without reference frame alignment: Expoiting photon orbital angular momentum [J]. Optics Communications, 2006, 260: 340-346
[48] C. Archer. There is no generalization of known formulas for mutually unbiased bases [J]. Phys. Rev. Lett., 2005, 46: 022-106
[49] P. Sulc, J. Tolar. Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions [J]. Phys. Rev. Lett., 2007, 40: 15099-15111
[50] M. Bourennane et al. Quantum key distribution using multi level encoding: security analysis [J]. Phys. Rev. Lett., 2002, 10065(35)
[51] M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani. Experimental realization of any discrete unitary operator [J]. Phys. Rev. Lett., 1994, 73: 58-61
[52] I. Csisz, J. Korner. Broadcast channels with confidential messages [J]. IEEE Trans. Inf. Theory, 24, 1978, 339-348
[53] M. J. W. Hall. Information exclusion: principle for complement observables [J]. Phys. Rev. Lett., 1995, 74: 3307-3310
[54] D. Mayers. Unconditional security in quantum cryptography [J]. Comput. Mac, 2001, 48: 351-406