基于双模波导中光场横向模式的量子纠缠的经典模拟
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摘要
量子纠缠是量子力学中最基本也是最独特的性质之一,由于它是实现量子计算和量子隐形传态的关键,在国际上得到了广泛关注和深入研究。
本文首先介绍了以量子计算为代表的量子信息学,量子纠缠的基本性质以及量子纠缠在量子信息学中的重要性。然后介绍了一种基于双模波导中光场横向模式的量子计算方案,该方案将波导中光场的横向模式编码为量子比特,并基于非线性光学实现了控制非门。在此基础上,本文提出了一种利用光场横向模式实现量子纠缠的经典模拟的思想,利用控制非门或线性光学器件实现的模式纠缠态作为量子纠缠态的经典模拟,并对这种模式纠缠态的性质进行了研究。这篇论文主要的工作如下:
首先,利用有限差分光束传播方法(FD-BPM)对基于非线性光学实现模式纠缠态的方案进行计算机模拟,通过基于Bell不等式的相关测量,得到模式纠缠态中两个场的相关函数,这个相关函数能够对Bell不等式产生破坏,证明模式纠缠态中的两个光场之间存在一种非局域的关联,可以作为量子纠缠态的经典模拟。进一步,本文建立了一个由随机相位差标定的系综模型来对模式纠缠态进行描述。
然后本文提出一种全新的基于线性光学的方案来实现两个光场的模式纠缠态。对这个方案进行的计算机模拟结果表明,这种模式纠缠态可以产生和量子纠缠态完全类似的相关函数,同样能够对Bell不等式产生破坏。这种方案更清晰地说明随机相位系综能够对模式纠缠态所显示的非局域性质进行准确的刻画,非局域性质很有可能就来源于这种随机相位机制。
接下来对线性光学方案进行推广,利用三个光场实现三粒子GHZ态的经典模拟。由GHZ定理可知,三粒子的完全关联可以给出量子纠缠存在的非统计判据,这种判据比基于统计性质的Bell不等式更为可靠。进一步的计算机模拟结果显示三个经典光场实现的模式纠缠态的相关函数与量子GHZ态完全类似,并且得到了模式纠缠态中非局域关联的非统计判据。这进一步证明了利用波导中横向模式来实现量子纠缠的可行性。
本文提出的实现量子纠缠的经典模拟方案在现有的实验条件下是完全可以实现的,因此我们进行了一些实验方面的工作,为进一步的实验验证作准备。文章的结尾给出了讨论和总结。
Quantum entanglement is one of the most fundamental and most special properties of quantum mechanics;it attracts wide attentions and in-depth researches due to the importance of quantum entanglement in quantum information.In this paper, we first introduce quantum information, the properties of quantum entanglement and the importance of quantum entanglement in quantum information. Then we introduce a quantum computation scheme based on optical transverse modes in multimode waveguide. In the scheme, quantum bits are encoded by using optical transverse modes, and CNOT gate is implemented by using nonlinear optics. Based on this, we propose a new conception of the so-called "mode-entangled states" to realize the classical simulation of quantum entangled states, which can be produced by CNOT gate or linear optical components. Further, we study some properties of the mode-entangled states. The main work of this paper is as follow:First, we simulate the nonlinear optical scheme using finite difference beam propagation method, and obtain correlation function of the two fields in mode entangled states based on correlation measurement of Bell inequality, the correlation function violate the Bell inequality, which proves there is nonlocal correlation between the two fields. Thus, the mode entangled states can be regarded as the classical simulation of the quantum entangled states. Further, we use an ensemble labeled by phase difference to describe mode entangled states.After that, we proposed a new scheme based on linear optics to realize mode entangled states. The numerical simulation results show that the correlation function of mode entangled states is quite similar to quantum entangled states, which can violate the Bell inequality. This scheme has proved that the phase difference ensemble can be used to describe the nonlocal correlation, and the nonlocal properties may originate from the random phase mechanism.And then we generalize the linear optical scheme to realize the GHZ states of three particles using three classical fields. According to GHZ theorem, the full correlation of three particles can give the nonstatistical criterion of quantum entanglement, which is more reliable than Bell inequality, which is based on statistical property. Further, we obtain the nonstatistical criterion of nonlocal correlation of mode entangled states, which again prove the validity of using classical fields to realize the classical simulation of quantum entanglement.The nonlinear and linear optical schemes proposed in the paper can be carried out in current condition;hence we have some experimental research to prepare the experimental verification. At last we give some discussion and conclusion.
本文首先介绍了以量子计算为代表的量子信息学,量子纠缠的基本性质以及量子纠缠在量子信息学中的重要性。然后介绍了一种基于双模波导中光场横向模式的量子计算方案,该方案将波导中光场的横向模式编码为量子比特,并基于非线性光学实现了控制非门。在此基础上,本文提出了一种利用光场横向模式实现量子纠缠的经典模拟的思想,利用控制非门或线性光学器件实现的模式纠缠态作为量子纠缠态的经典模拟,并对这种模式纠缠态的性质进行了研究。这篇论文主要的工作如下:
首先,利用有限差分光束传播方法(FD-BPM)对基于非线性光学实现模式纠缠态的方案进行计算机模拟,通过基于Bell不等式的相关测量,得到模式纠缠态中两个场的相关函数,这个相关函数能够对Bell不等式产生破坏,证明模式纠缠态中的两个光场之间存在一种非局域的关联,可以作为量子纠缠态的经典模拟。进一步,本文建立了一个由随机相位差标定的系综模型来对模式纠缠态进行描述。
然后本文提出一种全新的基于线性光学的方案来实现两个光场的模式纠缠态。对这个方案进行的计算机模拟结果表明,这种模式纠缠态可以产生和量子纠缠态完全类似的相关函数,同样能够对Bell不等式产生破坏。这种方案更清晰地说明随机相位系综能够对模式纠缠态所显示的非局域性质进行准确的刻画,非局域性质很有可能就来源于这种随机相位机制。
接下来对线性光学方案进行推广,利用三个光场实现三粒子GHZ态的经典模拟。由GHZ定理可知,三粒子的完全关联可以给出量子纠缠存在的非统计判据,这种判据比基于统计性质的Bell不等式更为可靠。进一步的计算机模拟结果显示三个经典光场实现的模式纠缠态的相关函数与量子GHZ态完全类似,并且得到了模式纠缠态中非局域关联的非统计判据。这进一步证明了利用波导中横向模式来实现量子纠缠的可行性。
本文提出的实现量子纠缠的经典模拟方案在现有的实验条件下是完全可以实现的,因此我们进行了一些实验方面的工作,为进一步的实验验证作准备。文章的结尾给出了讨论和总结。
Quantum entanglement is one of the most fundamental and most special properties of quantum mechanics;it attracts wide attentions and in-depth researches due to the importance of quantum entanglement in quantum information.In this paper, we first introduce quantum information, the properties of quantum entanglement and the importance of quantum entanglement in quantum information. Then we introduce a quantum computation scheme based on optical transverse modes in multimode waveguide. In the scheme, quantum bits are encoded by using optical transverse modes, and CNOT gate is implemented by using nonlinear optics. Based on this, we propose a new conception of the so-called "mode-entangled states" to realize the classical simulation of quantum entangled states, which can be produced by CNOT gate or linear optical components. Further, we study some properties of the mode-entangled states. The main work of this paper is as follow:First, we simulate the nonlinear optical scheme using finite difference beam propagation method, and obtain correlation function of the two fields in mode entangled states based on correlation measurement of Bell inequality, the correlation function violate the Bell inequality, which proves there is nonlocal correlation between the two fields. Thus, the mode entangled states can be regarded as the classical simulation of the quantum entangled states. Further, we use an ensemble labeled by phase difference to describe mode entangled states.After that, we proposed a new scheme based on linear optics to realize mode entangled states. The numerical simulation results show that the correlation function of mode entangled states is quite similar to quantum entangled states, which can violate the Bell inequality. This scheme has proved that the phase difference ensemble can be used to describe the nonlocal correlation, and the nonlocal properties may originate from the random phase mechanism.And then we generalize the linear optical scheme to realize the GHZ states of three particles using three classical fields. According to GHZ theorem, the full correlation of three particles can give the nonstatistical criterion of quantum entanglement, which is more reliable than Bell inequality, which is based on statistical property. Further, we obtain the nonstatistical criterion of nonlocal correlation of mode entangled states, which again prove the validity of using classical fields to realize the classical simulation of quantum entanglement.The nonlinear and linear optical schemes proposed in the paper can be carried out in current condition;hence we have some experimental research to prepare the experimental verification. At last we give some discussion and conclusion.
引文
[1] M. A. Nielsen and I. L. Chuang, "Quantum Computation and Quantum Information", Cambridge, Cambridge University Press, 2000
[2] C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation", Nature 404 (2000), 247
[3] R. P. Feynman, "Simulating Physics With Computers", Int. J. Theor. Phys. 21 (1982), 467
[4] D. Deutsch, "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer", Pro. Royal Soc. London. Series A 400 (1985), 97
[5] C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, (IEEE, New York, 1984), 175
[6] A. K. Ekert, "Quantum Cryptography Based on Bell's Theorem", Phys. Rev. Lett. 67 (1991), 661
[7] P. W. Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", Proceedings of Foundations of Computer Science, 124
[8] L. K. Grover, "A Fast Quantum Mechanical Algorithm for Database Search", quant-ph/9605043
[9] M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood and I. L. Chuang, "Experimental Realization of Shor's Quantum Factoring Algorithm Using Nuclear Magnetic Resonance", Nature 414 (2001), 883
[10] C. H. Bennett et al., "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels", Phys. Rev. Lett. 70 (1993), 1895
[11] D. Bouwmeester et al., "Experimental quantum teleportation", Nature 390 (1997), 575
[12] Z. Zhao et al., "Experimental demonstration of five-photon entanglement and open-destination teleportation", Nature 430 (2004), 54
[13] Y. F. Huang et al., "Experimental Teleportation of a Quantum Controlled-NOT Gate", Phys. Rev. Lett. 93 (1991), 240501
[14] X. F. Qian et al., "Quantum-state transfer characterized by mode entanglement", Phys. Rev. A 72(2005), 062329
[15] J. I. Cirac and P. Zoller, "Quantum Computations with Cold Trapped Ions", Phys. Rev. Lett. 74 (1995), 4091
[16] A. Barenco et al., "Conditional Quantum Dynamics and Logic Gates", Phys. Rev. Lett. 74 (1995), 4083
[17] T. Sleator et al., "Realization Universal Quantum Logic Gates", Phys. Rev. Lett. 74 (1995), 4087
[18] S. L. Braunstein et al., "Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing", Phys. Rev. Lett. 83 (1999), 1054
[19] D. Loss et al., "Quantum Computation with Quantum Dots", Phys. Rev. A 57 (1998), 120
[20] A. Wallraff, "Strong Coupling of a Single Photon to a Superconducting Qubit Using Circuit Quantum Electrodynamics", Nature 431 (2004), 162
[21] C. Mochon, "Anyons from nonsolvable finite groups are sufficient for universal quantum computation", Phys. Rev. A 67 (2003), 022315
[22] E. Bernstein, U. Vazirani, "Quantum complexity theory", SIAM Journal of Computing 11 (1993)
[23] D. R. Simon, "On the power of quantum computation", In Proc. 35th Symposium on Foundations of Computer Science (FOCS) (1994), 116
[24] C. H. Bennett and G. Brassard, "Quantum cryptography: Public key distribution and coin tossing", Proceedings of IEEE International Conference on Computers (1984)
[25] C. H. Bennett, "Quantum cryptography using any two nonorthogonal states", Phys. Rev. Lett. 68 (1992), 3121
[26] A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Phys. Rev. 47 (1935), 777
[27] J. S. Bell, "On the Einstein-Podolsky-Rosen paradox", Physics 1 (1964), 195
[28] J. S. Bell, "On the Problem of Hidden Variables in Quantum Mechanics", Rev. Mod. Phys., 38 (1966), 447
[29] D. Kennedy and C. Norman, "What Don't We Know?" Science 309 (2005), 75
[30] C. Seife, "Do Deeper Principles Underlie Quantum Uncertainty and Nonlocality?" Science 309 (2005), 98
[31] J. F. Clauser et al., "Proposed Experiment to Test Local Hidden-Variable Theories", Phys. Rev. Lett. 23(1969), 880
[32] "Bell's Theorem, Quantum Theory, and Conceptions of the Universe," D. M. Greenberger et al., edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989)
[33] "Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories" L. Hardy, Phys. Rev. Lett. 68,2981 (1992)
[34] L. M. Duan et al., "Inseparability Criterion for Continuous Variable Systems", Phys. Rev. Lett. 84 (2000), 2722
[35] E. Shchukin and W. Vogel, "Inseparability Criteria for Continuous Bipartite Quantum States", Phys. Rev. Lett. 95 (2005), 230502
[36] D. Leibfried et al., "Creation of a six-atom 'Schrodinger cat' state", Nature 438 (2005), 639
[37] Haffner et al., "Scalable multiparticle entanglement of trapped ions", Nature 438 (2005), 643
[38] C. Kurtsiefer et al., "Quantum cryptography: A step towards global key distribution", Nature 419 (2002), 450
[39] C. Z. Peng et al., "Experimental Free-Space Distribution of Entangled Photon Pairs Over 13 km: Towards Satellite-Based Global Quantum Communication", Phys. Rev. Lett. 94 (2005), 150501
[40] D. A. Lidar, "Comment on 'Quantum waveguide array generator for performing Fourier transforms: Alternate route to quantum computing'", Appl. Phys. Lett. 80 (2002), 2419
[41] D. M. Greenberger, M. A. Home, A. Shimony, A. Zeilinger, Am. J. Phys., 1990, 58 1131
[42] R. Jozsa et al., "On the role of entanglement in quantum computational speed-up", quant-ph/0201143
[43] E. Knill, R. Laflamme and G J. Milburn, "A scheme for efficient quantum computation with linear optics", Nature 409 (2001), 46
[44] J. D. Franson, et al, "Quantum logic operations based on photon-exchange interactions", Phys. Rev. A 60(1999),917
[45] S. Glancy, et al, "Implementation of a quantum phase gate by the optical Kerr effect", quant-ph/0009110
[46] R. G Beausoleil et al., "Applications of Coherent Population Transfer to Quantum Information Processing", quant-ph/0302109
[47] L. M. Kuang, G. H. Chen and Y. S. Wu, "Giant non-linearities accompanying electromagnetically induced transparency", quant-ph/0103152
[48] S. Lloyd, et al., "Quantum Computation over Continuous Variables", Phys. Rev. Lett. 82 (1999), 1784
[49] S. L. Lomonaco, "A Continuous Variable Shor Algorithm", quant-ph/0210141
[50] A. K. Pati et al, "Deutsch-Jozsa algorithm for continuous variables", quant-ph/0207108
[51] S. D. Bartlett et al, "Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting", Phys. Rev. A 65 (2002), 042304
[52] T. C. Ralph et al, "Quantum computation with optical coherent states", quant-ph/0306004
[53] A. Gilchrist et al, "Schrodinger cats and their power for quantum information processing", quant-ph/0312194
[54] K. F. Lee and J. E. Thomas, "Experimental Simulation of Two-Particle Quantum Entanglement using Classical Fields", Phys. Rev. Lett. (2002)88, 097902
[55] W. A. Hofer, "Numerical simulation of Einstein-Podolsky-Rosen experiments in a local hidden variables model", quant-ph/0108141
[56] W. A. Hofer, "Numerical simulation of interference experiments in a local hidden variables model", quant-ph/0111131
[57] J. Fu et al., "Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides", Phys. Rev. A 70 (2004), 042313
[58] J. E. Sipe, "Photon wave functions", Phys. Rev. A 52 (1995), 1875
[59] E. Wolf, Elsevier, Amsterdam, "Progress in Optics ⅩⅩⅩⅥ", (1996)
[60] D. H. Kobe, "A Relativistic Schrrdinger-like Equation for a Photon and Its Second Quantization", Found. Phys. 29 (1999), 1203
[61] G. Magyar and L. Mandel, "Interference Fringes Produced by Superposition of Two Independent Maser Light Beams", Nature 198 (1963), 255
[62] L. Mandel, "Quantum Theory of Interference Effects Produced by Independent Light Beams", Phys. Rev. 134 (1964), A10
[63] S. G. Krivoshlykov and I. N. Sissakian, "Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides", Opt. Quant. Elect. 12 (1980), 463
[64] D. Dragoman, "The Wigner distribution function in optics and optoelectronics", Prog. Opt. 42 (2002), 424
[65] K. F. Lee et al., "Heterodyne measurement of Wigner distributions for classical optical fields", Opt. Lett. 24 (1999), 1370
[66] C. C. Cheng and M. G. Raymer, "Long-Range Saturation of Spatial Decoherence in Wave-Field Transport in Random Multiple-Scattering Media", Phys. Rev. Lett. 82 (1999), 4807
[67] 张礼,葛墨林,量子力学的前沿问题,清华大学出版社,2000
[68] 吴重庆,光波导理论,清华大学出版社,2000
[2] C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation", Nature 404 (2000), 247
[3] R. P. Feynman, "Simulating Physics With Computers", Int. J. Theor. Phys. 21 (1982), 467
[4] D. Deutsch, "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer", Pro. Royal Soc. London. Series A 400 (1985), 97
[5] C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, (IEEE, New York, 1984), 175
[6] A. K. Ekert, "Quantum Cryptography Based on Bell's Theorem", Phys. Rev. Lett. 67 (1991), 661
[7] P. W. Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", Proceedings of Foundations of Computer Science, 124
[8] L. K. Grover, "A Fast Quantum Mechanical Algorithm for Database Search", quant-ph/9605043
[9] M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood and I. L. Chuang, "Experimental Realization of Shor's Quantum Factoring Algorithm Using Nuclear Magnetic Resonance", Nature 414 (2001), 883
[10] C. H. Bennett et al., "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels", Phys. Rev. Lett. 70 (1993), 1895
[11] D. Bouwmeester et al., "Experimental quantum teleportation", Nature 390 (1997), 575
[12] Z. Zhao et al., "Experimental demonstration of five-photon entanglement and open-destination teleportation", Nature 430 (2004), 54
[13] Y. F. Huang et al., "Experimental Teleportation of a Quantum Controlled-NOT Gate", Phys. Rev. Lett. 93 (1991), 240501
[14] X. F. Qian et al., "Quantum-state transfer characterized by mode entanglement", Phys. Rev. A 72(2005), 062329
[15] J. I. Cirac and P. Zoller, "Quantum Computations with Cold Trapped Ions", Phys. Rev. Lett. 74 (1995), 4091
[16] A. Barenco et al., "Conditional Quantum Dynamics and Logic Gates", Phys. Rev. Lett. 74 (1995), 4083
[17] T. Sleator et al., "Realization Universal Quantum Logic Gates", Phys. Rev. Lett. 74 (1995), 4087
[18] S. L. Braunstein et al., "Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing", Phys. Rev. Lett. 83 (1999), 1054
[19] D. Loss et al., "Quantum Computation with Quantum Dots", Phys. Rev. A 57 (1998), 120
[20] A. Wallraff, "Strong Coupling of a Single Photon to a Superconducting Qubit Using Circuit Quantum Electrodynamics", Nature 431 (2004), 162
[21] C. Mochon, "Anyons from nonsolvable finite groups are sufficient for universal quantum computation", Phys. Rev. A 67 (2003), 022315
[22] E. Bernstein, U. Vazirani, "Quantum complexity theory", SIAM Journal of Computing 11 (1993)
[23] D. R. Simon, "On the power of quantum computation", In Proc. 35th Symposium on Foundations of Computer Science (FOCS) (1994), 116
[24] C. H. Bennett and G. Brassard, "Quantum cryptography: Public key distribution and coin tossing", Proceedings of IEEE International Conference on Computers (1984)
[25] C. H. Bennett, "Quantum cryptography using any two nonorthogonal states", Phys. Rev. Lett. 68 (1992), 3121
[26] A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Phys. Rev. 47 (1935), 777
[27] J. S. Bell, "On the Einstein-Podolsky-Rosen paradox", Physics 1 (1964), 195
[28] J. S. Bell, "On the Problem of Hidden Variables in Quantum Mechanics", Rev. Mod. Phys., 38 (1966), 447
[29] D. Kennedy and C. Norman, "What Don't We Know?" Science 309 (2005), 75
[30] C. Seife, "Do Deeper Principles Underlie Quantum Uncertainty and Nonlocality?" Science 309 (2005), 98
[31] J. F. Clauser et al., "Proposed Experiment to Test Local Hidden-Variable Theories", Phys. Rev. Lett. 23(1969), 880
[32] "Bell's Theorem, Quantum Theory, and Conceptions of the Universe," D. M. Greenberger et al., edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989)
[33] "Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories" L. Hardy, Phys. Rev. Lett. 68,2981 (1992)
[34] L. M. Duan et al., "Inseparability Criterion for Continuous Variable Systems", Phys. Rev. Lett. 84 (2000), 2722
[35] E. Shchukin and W. Vogel, "Inseparability Criteria for Continuous Bipartite Quantum States", Phys. Rev. Lett. 95 (2005), 230502
[36] D. Leibfried et al., "Creation of a six-atom 'Schrodinger cat' state", Nature 438 (2005), 639
[37] Haffner et al., "Scalable multiparticle entanglement of trapped ions", Nature 438 (2005), 643
[38] C. Kurtsiefer et al., "Quantum cryptography: A step towards global key distribution", Nature 419 (2002), 450
[39] C. Z. Peng et al., "Experimental Free-Space Distribution of Entangled Photon Pairs Over 13 km: Towards Satellite-Based Global Quantum Communication", Phys. Rev. Lett. 94 (2005), 150501
[40] D. A. Lidar, "Comment on 'Quantum waveguide array generator for performing Fourier transforms: Alternate route to quantum computing'", Appl. Phys. Lett. 80 (2002), 2419
[41] D. M. Greenberger, M. A. Home, A. Shimony, A. Zeilinger, Am. J. Phys., 1990, 58 1131
[42] R. Jozsa et al., "On the role of entanglement in quantum computational speed-up", quant-ph/0201143
[43] E. Knill, R. Laflamme and G J. Milburn, "A scheme for efficient quantum computation with linear optics", Nature 409 (2001), 46
[44] J. D. Franson, et al, "Quantum logic operations based on photon-exchange interactions", Phys. Rev. A 60(1999),917
[45] S. Glancy, et al, "Implementation of a quantum phase gate by the optical Kerr effect", quant-ph/0009110
[46] R. G Beausoleil et al., "Applications of Coherent Population Transfer to Quantum Information Processing", quant-ph/0302109
[47] L. M. Kuang, G. H. Chen and Y. S. Wu, "Giant non-linearities accompanying electromagnetically induced transparency", quant-ph/0103152
[48] S. Lloyd, et al., "Quantum Computation over Continuous Variables", Phys. Rev. Lett. 82 (1999), 1784
[49] S. L. Lomonaco, "A Continuous Variable Shor Algorithm", quant-ph/0210141
[50] A. K. Pati et al, "Deutsch-Jozsa algorithm for continuous variables", quant-ph/0207108
[51] S. D. Bartlett et al, "Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting", Phys. Rev. A 65 (2002), 042304
[52] T. C. Ralph et al, "Quantum computation with optical coherent states", quant-ph/0306004
[53] A. Gilchrist et al, "Schrodinger cats and their power for quantum information processing", quant-ph/0312194
[54] K. F. Lee and J. E. Thomas, "Experimental Simulation of Two-Particle Quantum Entanglement using Classical Fields", Phys. Rev. Lett. (2002)88, 097902
[55] W. A. Hofer, "Numerical simulation of Einstein-Podolsky-Rosen experiments in a local hidden variables model", quant-ph/0108141
[56] W. A. Hofer, "Numerical simulation of interference experiments in a local hidden variables model", quant-ph/0111131
[57] J. Fu et al., "Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides", Phys. Rev. A 70 (2004), 042313
[58] J. E. Sipe, "Photon wave functions", Phys. Rev. A 52 (1995), 1875
[59] E. Wolf, Elsevier, Amsterdam, "Progress in Optics ⅩⅩⅩⅥ", (1996)
[60] D. H. Kobe, "A Relativistic Schrrdinger-like Equation for a Photon and Its Second Quantization", Found. Phys. 29 (1999), 1203
[61] G. Magyar and L. Mandel, "Interference Fringes Produced by Superposition of Two Independent Maser Light Beams", Nature 198 (1963), 255
[62] L. Mandel, "Quantum Theory of Interference Effects Produced by Independent Light Beams", Phys. Rev. 134 (1964), A10
[63] S. G. Krivoshlykov and I. N. Sissakian, "Optical beam and pulse propagation in inhomogeneous media. Application to multimode parabolic-index waveguides", Opt. Quant. Elect. 12 (1980), 463
[64] D. Dragoman, "The Wigner distribution function in optics and optoelectronics", Prog. Opt. 42 (2002), 424
[65] K. F. Lee et al., "Heterodyne measurement of Wigner distributions for classical optical fields", Opt. Lett. 24 (1999), 1370
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