连续变量纠缠态表象在量子测量和热场动力学中的应用
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摘要
量子纠缠是量子信息科学与技术的核心资源,如果能用纠缠态表象来描述量子纠缠,就可以使量子纠缠的阐述更为清晰,通过建立连续变量纠缠态表象就会给量子测量带来很大的便利。由于自然界中绝大多数系统都是“浸”在一个热库中,所以系统─热库构成的总系统中就会存在着某种量子纠缠。探讨连续变量纠缠态表象在量子测量和热场动力学中的应用,就是本文要讨论的主要内容。
我们用纠缠态表象方法计算了对双模压缩真空态第一个模场施行正交振幅测量,结果发现,第二个模塌缩到一个压缩趋强的压缩态,导出了它的显函数表达式。
利用纠缠态表象,我们对具有两个压缩参数的广义双模压缩态作单模正交振幅的测量所引起的另一态的塌缩结果作了理论分析,结果表明,测量后第二个模塌缩为一个与这两个压缩参数都有关的单模压缩态。当用单模压缩态投影算符作为测量基,对三参量SU(1,1)相干态施行测量,另一个场模变为一个单模压缩态,这个压缩态不仅与SU(1,1)参数有关,而且与测量算符的压缩参数有关。这个结果呈现带压缩的量子纠缠,意味着测量模可以来控制光场模,从而达到构建新量子态的目的。
根据双模压缩数态是双模压缩真空态的双变量厄米多项式激发态,得到了单模l ?光子数投影算符对双模压缩数态S_2|m, n>施行测量的结果。结果表明,剩余的场模塌缩到数态|n - m + l>,其系数是是n ,m和l的雅可比多项式,表现了两模之间的纠缠,即塌缩态依赖于两个模之间的数差。当第一个模是用相干态去测量时,第二个模则塌缩到一种激发相干态。这些结果丰富了利用双模压缩态实现量子隐形传态的理论。
通过分析热场动力学的特性,我们引入了在热平衡状态下一个新的态|z,N,>它是热不变相干态,这个态不仅能保持系统和热库的总能量不变,而且还是一个广义相干态。当整个系统处在这个态时,湮灭系统的一个量子,同时湮灭热库的一个有负能的空穴将不会改变系统的总能量。
利用热Wigner算符在热场纠缠态表象中的表达式和有序算符内的积分技术,我们导出了热不变相干态|z,N>的Wigner函数,利用Wigner函数的负值讨论了热不变相干态的非经典性质。结果表明,热不变相干态|z,N>的非经典性质与相干态的参量z和热平衡时整个系统的总能量都有关系。当取不同的值时,态|z,N>表现出不同的非经典性质。特别是当总能量为奇数时,态|z,N>的非经典性质更明显。
在热场动力学理论中,对每一个真实的光学场模须引进一个“虚拟”的场模,所以光子数态应为|n,(n|ˇ)>。对于包含热库效应在内的总系统的哈密顿算符存在h |n,(n|ˇ)> = 0,但它不能体现本征值是n ,对描述一定温度下的热激发态是不方便的。为了避免态|n,(n|ˇ)>的不足,通过引入合适的热激发算符,我们找到了热场动力学理论中新的热激发态,它是哈密顿量h具有本征值D的真正的本征矢。态|D ,n>不仅能体现h的本征值为D,还能表现出热激发行为,是一个热激发态。值得注意的是它也是纠缠态。最后,我们给出了新的热激发态在构造新热压缩态和相态中的可能应用。本文所提出的热不变相干态和新的热激发态丰富了热场动力学理论。
The quantum entanglement is a core resource in quantum information science. If the quantum entanglement can be represented by the entangled state representation, the quantum entanglement can be more clearly expounded. The establishment of continuous variable entangled state representation can bring great convenience to quantum measurement. Since most realistic systems are immersed in a“thermal reservoir”, there is a certain quantum entanglement in the total system of the subsystem and the reservoir. Some applications of continuous variable entangled state representation in the quantum measurement and in the thermo field dynamics (TFD) theory are the major discussions of the paper.
We calculate the result of the quadrature-amplitude measurement on the first-mode field of a two-mode squeezed vacuum state by virtue of the entangled state representation. It is found that the second-mode field collapses to a single-mode squeezed state with stronger squeezing. The explicit form is derived.
By virtue of the entangled state representation, we analyze the result of one- mode quadrature-amplitude measurement for some generalized two-mode squeezed states with two squeezing parameters. It is found that the remaining field-mode simultaneously collapses to the single-mode squeezed state, which is related to both squeezing parameters. When a measurement for the 3-parameter SU(1,1) squeezing state is performed with the first-mode field in the single-mode squeezed state, the second-mode field becomes a single-mode squeezed state. Its squeezing depends not only on the SU(1,1) parameter, but also on the squeezing parameter of the measurement operator. This shows the quantum entanglement with squeezing. That is, the light field mode can be controlled by the measurement mode, so as to establish a new quantum state.
Based on the fact that the two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l-photon measurement for the two-mode squeezed number states S 2 m, n is obtained. It is found that the remaining field-mode simultaneously collapses to the number state n ? m + l with the coefficient being a Jacobi polynomial of n ,m and l . It manifestly exhibits the entanglement between the two modes, i. e., it depends on the number-difference between the two modes. The second mode collapses to an excited coherent state when the first mode is measured as a coherent state.Therefore, the quantum teleportation theory can be further developed by these results.
By analyzing the characteristics of thermo field dynamics, a new physical state |z,N> at thermal equilibrium has been found. It is the thermo-invariant coherent state. This state not only maintains the whole energy of the system and the reservoir but also is a generalized coherent state. When the whole system is in this state, the energy of the whole system will not change by simultaneously annihilating a quantum of the system and a hole with negative energy in the reservoir.
By using thermal Winger operator of the TFD in the coherent thermal state representation and the integration technique in an ordered product of operators, the Wigner function of the thermo-invariant coherent state |z,N> is derived. The nonclassical property of the state |z,N> is discussed based on the negativity of the Wigner function. It is found that the nonclassical statistical properties of the thermo- invariant coherent state |z,N> depend on both the parameter z of the coherent state and the energy N of the whole system in thermal equilibrium. For different values of N, the state |z,N> can exhibit different nonclassical statistical properties. The nonclassicality of the state |z,N> is more pronounced when N is an odd number.
In the thermo field dynamics, every real optical field mmode can be accompanied by a fictitious field mode. Therefore, the photon number state (n-excitation state) can be denoted by |n,(n|ˇ)> . The Hamiltonian of the total system including the effects of the reservoir holds for h |n,(n|ˇ)> = 0. It cannot embody the eigenvalue n. It is inconvenient to describe thermal excitation of the system at finite temperature. To avoid the weakness of |n,(n|ˇ)> , new thermo excitation states are successfully constructed by introducing appropriate thermo excitation operators. It is the eigenvector of the Hamiltonian h with the eigenvalue D. The state ||D ,n> can exhibit not only the eigenvalue D of h, but also the thermal excitation behaviour. It is noted that it is a thermal excitation state and an entangled state. Applications of the new thermo excitation state in constructing new thermo squeezed state and phase state are also presented. The TFD theory can be further developed by the thermo- invariant coherent state and the new thermo excitation state.
我们用纠缠态表象方法计算了对双模压缩真空态第一个模场施行正交振幅测量,结果发现,第二个模塌缩到一个压缩趋强的压缩态,导出了它的显函数表达式。
利用纠缠态表象,我们对具有两个压缩参数的广义双模压缩态作单模正交振幅的测量所引起的另一态的塌缩结果作了理论分析,结果表明,测量后第二个模塌缩为一个与这两个压缩参数都有关的单模压缩态。当用单模压缩态投影算符作为测量基,对三参量SU(1,1)相干态施行测量,另一个场模变为一个单模压缩态,这个压缩态不仅与SU(1,1)参数有关,而且与测量算符的压缩参数有关。这个结果呈现带压缩的量子纠缠,意味着测量模可以来控制光场模,从而达到构建新量子态的目的。
根据双模压缩数态是双模压缩真空态的双变量厄米多项式激发态,得到了单模l ?光子数投影算符对双模压缩数态S_2|m, n>施行测量的结果。结果表明,剩余的场模塌缩到数态|n - m + l>,其系数是是n ,m和l的雅可比多项式,表现了两模之间的纠缠,即塌缩态依赖于两个模之间的数差。当第一个模是用相干态去测量时,第二个模则塌缩到一种激发相干态。这些结果丰富了利用双模压缩态实现量子隐形传态的理论。
通过分析热场动力学的特性,我们引入了在热平衡状态下一个新的态|z,N,>它是热不变相干态,这个态不仅能保持系统和热库的总能量不变,而且还是一个广义相干态。当整个系统处在这个态时,湮灭系统的一个量子,同时湮灭热库的一个有负能的空穴将不会改变系统的总能量。
利用热Wigner算符在热场纠缠态表象中的表达式和有序算符内的积分技术,我们导出了热不变相干态|z,N>的Wigner函数,利用Wigner函数的负值讨论了热不变相干态的非经典性质。结果表明,热不变相干态|z,N>的非经典性质与相干态的参量z和热平衡时整个系统的总能量都有关系。当取不同的值时,态|z,N>表现出不同的非经典性质。特别是当总能量为奇数时,态|z,N>的非经典性质更明显。
在热场动力学理论中,对每一个真实的光学场模须引进一个“虚拟”的场模,所以光子数态应为|n,(n|ˇ)>。对于包含热库效应在内的总系统的哈密顿算符存在h |n,(n|ˇ)> = 0,但它不能体现本征值是n ,对描述一定温度下的热激发态是不方便的。为了避免态|n,(n|ˇ)>的不足,通过引入合适的热激发算符,我们找到了热场动力学理论中新的热激发态,它是哈密顿量h具有本征值D的真正的本征矢。态|D ,n>不仅能体现h的本征值为D,还能表现出热激发行为,是一个热激发态。值得注意的是它也是纠缠态。最后,我们给出了新的热激发态在构造新热压缩态和相态中的可能应用。本文所提出的热不变相干态和新的热激发态丰富了热场动力学理论。
The quantum entanglement is a core resource in quantum information science. If the quantum entanglement can be represented by the entangled state representation, the quantum entanglement can be more clearly expounded. The establishment of continuous variable entangled state representation can bring great convenience to quantum measurement. Since most realistic systems are immersed in a“thermal reservoir”, there is a certain quantum entanglement in the total system of the subsystem and the reservoir. Some applications of continuous variable entangled state representation in the quantum measurement and in the thermo field dynamics (TFD) theory are the major discussions of the paper.
We calculate the result of the quadrature-amplitude measurement on the first-mode field of a two-mode squeezed vacuum state by virtue of the entangled state representation. It is found that the second-mode field collapses to a single-mode squeezed state with stronger squeezing. The explicit form is derived.
By virtue of the entangled state representation, we analyze the result of one- mode quadrature-amplitude measurement for some generalized two-mode squeezed states with two squeezing parameters. It is found that the remaining field-mode simultaneously collapses to the single-mode squeezed state, which is related to both squeezing parameters. When a measurement for the 3-parameter SU(1,1) squeezing state is performed with the first-mode field in the single-mode squeezed state, the second-mode field becomes a single-mode squeezed state. Its squeezing depends not only on the SU(1,1) parameter, but also on the squeezing parameter of the measurement operator. This shows the quantum entanglement with squeezing. That is, the light field mode can be controlled by the measurement mode, so as to establish a new quantum state.
Based on the fact that the two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l-photon measurement for the two-mode squeezed number states S 2 m, n is obtained. It is found that the remaining field-mode simultaneously collapses to the number state n ? m + l with the coefficient being a Jacobi polynomial of n ,m and l . It manifestly exhibits the entanglement between the two modes, i. e., it depends on the number-difference between the two modes. The second mode collapses to an excited coherent state when the first mode is measured as a coherent state.Therefore, the quantum teleportation theory can be further developed by these results.
By analyzing the characteristics of thermo field dynamics, a new physical state |z,N> at thermal equilibrium has been found. It is the thermo-invariant coherent state. This state not only maintains the whole energy of the system and the reservoir but also is a generalized coherent state. When the whole system is in this state, the energy of the whole system will not change by simultaneously annihilating a quantum of the system and a hole with negative energy in the reservoir.
By using thermal Winger operator of the TFD in the coherent thermal state representation and the integration technique in an ordered product of operators, the Wigner function of the thermo-invariant coherent state |z,N> is derived. The nonclassical property of the state |z,N> is discussed based on the negativity of the Wigner function. It is found that the nonclassical statistical properties of the thermo- invariant coherent state |z,N> depend on both the parameter z of the coherent state and the energy N of the whole system in thermal equilibrium. For different values of N, the state |z,N> can exhibit different nonclassical statistical properties. The nonclassicality of the state |z,N> is more pronounced when N is an odd number.
In the thermo field dynamics, every real optical field mmode can be accompanied by a fictitious field mode. Therefore, the photon number state (n-excitation state) can be denoted by |n,(n|ˇ)> . The Hamiltonian of the total system including the effects of the reservoir holds for h |n,(n|ˇ)> = 0. It cannot embody the eigenvalue n. It is inconvenient to describe thermal excitation of the system at finite temperature. To avoid the weakness of |n,(n|ˇ)> , new thermo excitation states are successfully constructed by introducing appropriate thermo excitation operators. It is the eigenvector of the Hamiltonian h with the eigenvalue D. The state ||D ,n> can exhibit not only the eigenvalue D of h, but also the thermal excitation behaviour. It is noted that it is a thermal excitation state and an entangled state. Applications of the new thermo excitation state in constructing new thermo squeezed state and phase state are also presented. The TFD theory can be further developed by the thermo- invariant coherent state and the new thermo excitation state.
引文
[1]李承祖等,“量子通信与量子计算”,国防科技大学出版社,2000
[2] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classic and Einstein-Podolsky-Rose channels, Phys. Rev. Lett. 70 (1993) 1895
[3] D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurther, and A. Zeilinger, Expermental quantum teleportation, Nature, 390 (1997) 575
[4] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popesu, Ladder proof of nonlocality without inequalities: theoretical and experimental results, Phys. Red. Lett. 80 (1998) 1121
[5] M. A. Nielsen, E. Knill, and R. Laflamme, Complete quantum teleportation using nuclear magnetic resonance, Nature 396 (1998) 52
[6] Y. H. Kim, S. P. Kulik, and Y. H. Shih, Quantum teleportation of a polarization state with a complete Bell state measurement, Phys. Rev. Lett. 86 (2001) 1370
[7] E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, Teleportation of a vacuum-one-photon qubit, Phys. Rev. Lett. 88 (2002) 070402
[8] L. Braunstein and H. J. Kimble, Teleportation of Continuous Quantum Variables Phys. Rev. Lett. 80 (1998) 869
[9] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000
[10] S. L. Braunstein and A. K. Pati, Quantum Information with Continuous Variables,Kluwer Academic Publishers, Dordrecht, 2003
[11] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[12] Memorial issue for H. Umezawa, Int. J. Phys. B 10 (1996) and reference therein
[13] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[14] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam) 1982
[15] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[16] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[17] A. Mann, M. Revzen, Thermal noise from pure state quantum correlations, Phys. Rev. A 40 (1989) 1674
[18] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[19] M. Berman, Thermal quasiparticles with phase-space distribution, Phys. Rev. A 40 (1989) 2057
[20] A. Mann, K. Nakamura, M. Revzen, H. Umezawa and Y. Yamanaka, Coherent and Thermal Coherent State, Jour. Math. Phys. 30 (1989) 2583
[21] A. Einstein, B. Podolsky and N. Rosen,“Can quantum-mechanical description of physical reality be consideres xomplete?”, Phys. Rev. 47 (1935) 777
[22] H. Y. Fan, J. R. Klauder, Eigenvectors of two particles\' relative position and total momentum, Phys. Rev. A 49 (1994) 704
[23] H. Y. Fan,Ye Xiong, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum Phys. Rev. A51 (1995) 3343
[24] P. G. Kwiat, H. Weinfuter, T. Herzog, and A. Zeilinger, New High-Intensity Source of Polarization-Entangled Photon Pairs, Phys. Rev. Lett. 75 (1995) 4337
[25] Z. Y. Ou, S. F. Pereira, H. J. Kimble, K. C. Peng, Realization of the Einstein- Podolsky-Rosen Paradox for Continuous Variables, Phys. Rev. Lett. 68 (1992) 3663
[26] Y. Zhang, H. Wang, X. ying, J. T. Jing, C. D. Xie, K. C. Peng, Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier, Phys. Rev. A 62 (2000) 023813
[27] X. Y. Li, Q. Pan, J. T. Jing, J. Zhang, C. D. Xie, and K. C. Peng, Quantum Dense Coding Exploiting a Bright Einstein-Podolsky-Rosen Beam, Phys. Rev. Lett. 88, (2000) 047904
[28] A. Furusawa, J. L. Serensen, S. L. Braunstein et al. Unconditional Quantum Teleportation Science 282, (1998) 706
[29] G. Leuchs, T. C. Ralph, Ch. Silberhorn, and N. Korolkova, Scheme for the generation of entangled solitons for quantum communication, J. Mod. Opt. 46 (1999) 1927
[30] Ch. Silberhorn, P. K. Lam, O. Weiβ, F. K?nig, N. Korolkova, and G. Leuchs, Generation of Continuous Variable Einstein-Podolsky-Rosen Entanglement via the Kerr Nonlinearity in an Optical Fiber, Phys. Rev. Lett. 86 (2001) 4267
[31] J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, K. Peng, Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables, Phys. Rev. Lett. 90 (2003) 167903
[32] T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, and A. Furusawa, Experimental Creation of a Fully Inseparable Tripartite Continuous-Variable State, Phys. Rev. Lett. 91 (2003) 080404
[33] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, Tripartite Quantum State Sharing, Arxiv: quant-ph/0311015 (2003)
[34] P. A. M. Dirac, The Principle of Quantum Mechanics, 3rd, Clarendon Press, Oxford 1930
[35] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[36] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[37] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[38] A. M. Perelomov, Generalized Coherent states and Their Application. Berlin: Springer-Verlag 1986
[39] V. Fock Zeits Für, Verallgemeinerung und L?sung der Diracschen statistischen Gleichung, Phys. 49 (1928) 339
[40] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D35 (1987) 1831
[41] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[42] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A 37 (1988) 2985
[43] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[44] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[45] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J.Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[46] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, (1997)
[47] W. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1974
[48] H. Y. Fan, Entangled states, squeezed states gained via the route of developing dirac’s symbolic method and their applications, Inter. J. Mod. Phys. 18 (2004) 1387
[49] H. P. Yuen, Two—photon coherent states of the radiation field, Phys. Rev. A 13 (1976) 2226
[50] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987)
[51] D. F. Walls, Squeezed states of light, Nature 306 (1983) 141
[52] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 145
[53] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum,Phys. Rev. A 49 (1994) 704
[54] H. Y. Fan and Chen Bo-zhan, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[55] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum, Phys. Rev. A 51 (1995) 3343
[56] H. Y. Fan, H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeezed operators, Phys. Rev. D 35 (1987) 1831
[57] H. Y. Fan and H. Zou, Similarity transformation operators as the images of classical symplectic transformations in coherent state representation, Phys. Lett. A 252 (1999) 281
[1] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[2] M. Orszg, Quantum Optics, Springer-Verlag, Berlin 2000
[3] P. S. Wolfgang, Quantum Optics in Phase Space, Wiley-Vch, Berlin, 2001
[4] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambidge University Press, 1977
[5] G. M. D'Ariano, M. G. Rassetti, J. Katriel, A. I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light (Plenum, New York,), 1989, p.301
[6] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[7] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D 35 (1987) 1831
[8] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[9] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A 37 (1988) 2985
[10] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[11] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[12] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[13] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 145
[14] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum, Phys. Rev. A 49 (1994) 704
[15] H. Y. Fan and B. Z. Chen, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[16] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum,Phys. Rev. A 51 (1995) 3343
[17] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be consideres xomplete?, Phys. Rev. 47 (1935) 777
[18] M. D. Reid, P. D. Drumond. Quantum correlation of phase in nondegenerate parametric oscilation, Phys. Rev. Lett. 69 (1989) 2731
[19] P. D. Drumond, M. D. Reid, Correlation in nondegenerate parametrix oscollation.Ⅱ.below threshold results, Phys. Rev. A 41 (1990) 3930.
[20] Z. Y. Ou, S. F. Pereira, H. J. Kimble and K. C. Peng, Realization of the Einstein-Podolsdy-Rosen for continuous variables, Phys Rev Lett. 68 3663-3666
[21] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[22] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[23] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[24] H. Y. Fan,Squeezed states: Operators for two types of one- and two-mode squeezing transformation, Phys. Rev. A 41 (1990) 1526
[25] H. Y. Fan,Application of entangled state representation in quantum teleportation with continuous variables, Phys. Lett. A 294 (2002) 253
[26] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers 1997
[1] A. Einstein, B. Podolsky and N. Rosen,“Can quantum-mechanical description of physical reality be consideres xomplete?”, Phys. Rev. 47 (1935) 777
[2] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[3] M. Orszg, Quantum Optics, Springer-Verlag, Berlin 2000
[4] P. S. Wolfgang, Quantum Optics in Phase Space, Wiley-Vch, Berlin, 2001
[5] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambidge University Press, 1977
[6] G. M. D’Ariano, M. G. Rassetti, J. Katriel and A. I. Solomon Squeezed and Nonclassical Light ed P. Tombesi and E. R. Pike, Nek York: Plenum, 1989
[7] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[8] H. X. Lu and B. Zhao, Relative entropy as a measure of entanglement for Gaussian states Chin. Phys. 15 (2006) 1914
[9] Y. B. Zhan, Controlled teleportation of high-dimension quantumstate with generalized bell-state measurement, Chin. Phys. 16 (2007) 2557
[10] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootter, Mixed- state entanglement and quant um error correction, Phys. Rev. A 54 (1996) 3824
[11] V. Vedral, M. B. Plenio, M. A. Rippin and P. L..Knight, Quantifying entanglement, Phys. Rev. Lett. 78 (1997) 2275
[12] M. B. Plenio ana V. Vedral, Bounds on relative entropy of entanglement for multi-party systems, J. Phys. A: Math. Gen. 34 (2001) 6997
[13] J. Eisert and H. J. Briegel, Schmidt measure as a tool for quantifying multiparticle entanglement, Phys. Rev. A 64 (2001) 022306
[14] T. C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68 (2003) 042307
[15] L. Tamaryan, D. K. Park, J. W. Son and S. Tamaryan Phys. Rev. A 78 (2008) 032304
[16] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 150
[17] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum,Phys. Rev. A 49 (1994) 704
[18] H. Y. Fan and B. Z. Chen, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[19] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum,Phys. Rev. A 51 (1995) 3343
[20] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[21] L. Y. Hu and H. Y. Fan, Two-mode squeezed number state as a two-variable Hermitepolynomial excitation on the squeezed vacuum, J. Mod. Opt. 55 (2008) 2011
[22] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[23] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[24] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[1] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[2] Memorial issue for H. Umezawa, Int. J. Phys. B10 (1996) and reference therein
[3] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[4] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam), 1982
[5] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[6] H. Y. Fan and Y. Fan, New representation of thermal states in thermal field dynamics, Phys. Lett. A246 (1998) 242
[7] H. Y. Fan and Y. Fan, New representation for thermo excitation and de-excitation in thermofield dynamics, Phys. Lett. A 269 (2001) 282
[8] H. Y. Fan, New application of thermo field dynamics in simplifying the calculation of wigner functions, Mod.Phys. Lett. A 18 (2003) 733
[9] H. Y. Fan, New Application of Thermo Field Dynamics in simplifying the calculation of Wigner functions, Mod. Phys. Lett. A, 18 (2003) 733
[10] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[11] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D35 (1987) 1831
[12] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[13] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A37 (1988) 2985
[14] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[15] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[16] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J.Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[17] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987)
[18] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[19] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[20] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[21] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[22] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[1] E. P. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40 (1932) 749
[2] R. F. O’Connel and E. P. Wigner, Quantum-Mechanical Distribution Functions: Conditions for Uniqueness, Phys. Lett. A 83 (1981) 145
[3] V. Bu?ek, C. H. Keitel. and P. L. Knight, Sampling entropies and operational phase-space measurement. II. Detection of quantum coherences, Phys. Rev. A 51 (1995) 2575
[4] R. P. Feynman Statistical Mechanics (New York:Benjamin) 1972
[5] K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Phys. Rev. A 40 (1989) 2847
[6] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators, Phys. Rev. D 2 (1970) 2161
[7] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space, Phys. Rev. D 2 (1970) 2187
[8] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. III. A Generalized Wick Theorem and Multitime Mapping, Phys. Rev. D 2 (1970) 2206
[9] D. F. Walls and G. Miburn, Quantum Optics (Berlin: Springer) 1994
[10] D. T. Smithey, M. Beck, A. Faridani, and M. G. Raymer, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum, Phys. Rev. Lett. 70 (1993) 1244
[11] H. Weyl, Quantenmechanik und Gruppentheorie Z. Phys. 46 (1927) 1-39
[12] M. G. Raymer, Contemporary Physics, 38 (1997)343
[13] K. E. Cahill and R. J. Glauber, Ordered Expansions in Boson Amplitude Operators, Phys. Rev. 177 (1969) 1857
[14] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co., 1985
[15] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[16] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[17] H. Y. Fan and H. R. Zaidi, Application of IWOP technique to the generalized Weyl correspondence, Phys. Lett. A 124(1987) 303
[18] H. Y. Fan and T.-N. Ruan, Coherent state formulation of the Weyl correspondence and the Wigner function, Commun. Theor. Phys. (Beijing, China) 2 (1983) 1563
[19] H. Y. Fan and T. N. Ruan, Quantum theory for fermion system and its pseudo-classical correspondence, Commun. Theor. Phys. (Beijing, China) 3 (1984) 45
[20] H. Weyl, Quantum mechanics and group theory, Z. Phys. 46 (1927) 1
[21] H. Y. Fan, Weyl-ordered operator moyal bracket by virtue of method of integral within a Weyl ordered product of operators, Commun. Theor. Phys. (Beijing, China) 3 (2006) 245
[22] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[1] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[2] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam), 1982
[3] Memorial issue for H. Umezawa, Int. J. Phys. B10 (1996) and reference therein
[4] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[5] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[6] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[7] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[8] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[9] G. M. D'Ariano, M. G. Rassetti, J. Katriel, A. I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light (Plenum, New York,), 1989, p.301
[10] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[11] H. Y. Fan and Y. Fan, New representation of thermal states in thermal field dynamics, Phys. Lett. A 246 (1998) 242
[12] H. Y. Fan and Fan Yue, New representation for thermo excitation and de-excitation in thermo field dynamics, Phys. Lett. A 269 (2001) 282
[13] H. Y. Fan, New application of thermo field dynamics in simplifying the calculation of wigner functions, Mod.Phys. Lett. A 18 (2003) 733
[14] H. Y. Fan, New Application of Thermo Field Dynamics in simplifying the calculation of Wigner functions, Mod. Phys. Lett. A, 18 (2003) 733
[15] H. Paul, Phase of a microscopic electromagnetic field and its measurement ,Fortschr. Phys. 22, 657 (1974)
[16] H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[2] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classic and Einstein-Podolsky-Rose channels, Phys. Rev. Lett. 70 (1993) 1895
[3] D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurther, and A. Zeilinger, Expermental quantum teleportation, Nature, 390 (1997) 575
[4] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popesu, Ladder proof of nonlocality without inequalities: theoretical and experimental results, Phys. Red. Lett. 80 (1998) 1121
[5] M. A. Nielsen, E. Knill, and R. Laflamme, Complete quantum teleportation using nuclear magnetic resonance, Nature 396 (1998) 52
[6] Y. H. Kim, S. P. Kulik, and Y. H. Shih, Quantum teleportation of a polarization state with a complete Bell state measurement, Phys. Rev. Lett. 86 (2001) 1370
[7] E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, Teleportation of a vacuum-one-photon qubit, Phys. Rev. Lett. 88 (2002) 070402
[8] L. Braunstein and H. J. Kimble, Teleportation of Continuous Quantum Variables Phys. Rev. Lett. 80 (1998) 869
[9] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000
[10] S. L. Braunstein and A. K. Pati, Quantum Information with Continuous Variables,Kluwer Academic Publishers, Dordrecht, 2003
[11] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[12] Memorial issue for H. Umezawa, Int. J. Phys. B 10 (1996) and reference therein
[13] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[14] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam) 1982
[15] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[16] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[17] A. Mann, M. Revzen, Thermal noise from pure state quantum correlations, Phys. Rev. A 40 (1989) 1674
[18] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[19] M. Berman, Thermal quasiparticles with phase-space distribution, Phys. Rev. A 40 (1989) 2057
[20] A. Mann, K. Nakamura, M. Revzen, H. Umezawa and Y. Yamanaka, Coherent and Thermal Coherent State, Jour. Math. Phys. 30 (1989) 2583
[21] A. Einstein, B. Podolsky and N. Rosen,“Can quantum-mechanical description of physical reality be consideres xomplete?”, Phys. Rev. 47 (1935) 777
[22] H. Y. Fan, J. R. Klauder, Eigenvectors of two particles\' relative position and total momentum, Phys. Rev. A 49 (1994) 704
[23] H. Y. Fan,Ye Xiong, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum Phys. Rev. A51 (1995) 3343
[24] P. G. Kwiat, H. Weinfuter, T. Herzog, and A. Zeilinger, New High-Intensity Source of Polarization-Entangled Photon Pairs, Phys. Rev. Lett. 75 (1995) 4337
[25] Z. Y. Ou, S. F. Pereira, H. J. Kimble, K. C. Peng, Realization of the Einstein- Podolsky-Rosen Paradox for Continuous Variables, Phys. Rev. Lett. 68 (1992) 3663
[26] Y. Zhang, H. Wang, X. ying, J. T. Jing, C. D. Xie, K. C. Peng, Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier, Phys. Rev. A 62 (2000) 023813
[27] X. Y. Li, Q. Pan, J. T. Jing, J. Zhang, C. D. Xie, and K. C. Peng, Quantum Dense Coding Exploiting a Bright Einstein-Podolsky-Rosen Beam, Phys. Rev. Lett. 88, (2000) 047904
[28] A. Furusawa, J. L. Serensen, S. L. Braunstein et al. Unconditional Quantum Teleportation Science 282, (1998) 706
[29] G. Leuchs, T. C. Ralph, Ch. Silberhorn, and N. Korolkova, Scheme for the generation of entangled solitons for quantum communication, J. Mod. Opt. 46 (1999) 1927
[30] Ch. Silberhorn, P. K. Lam, O. Weiβ, F. K?nig, N. Korolkova, and G. Leuchs, Generation of Continuous Variable Einstein-Podolsky-Rosen Entanglement via the Kerr Nonlinearity in an Optical Fiber, Phys. Rev. Lett. 86 (2001) 4267
[31] J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, K. Peng, Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables, Phys. Rev. Lett. 90 (2003) 167903
[32] T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, and A. Furusawa, Experimental Creation of a Fully Inseparable Tripartite Continuous-Variable State, Phys. Rev. Lett. 91 (2003) 080404
[33] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, Tripartite Quantum State Sharing, Arxiv: quant-ph/0311015 (2003)
[34] P. A. M. Dirac, The Principle of Quantum Mechanics, 3rd, Clarendon Press, Oxford 1930
[35] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[36] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[37] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[38] A. M. Perelomov, Generalized Coherent states and Their Application. Berlin: Springer-Verlag 1986
[39] V. Fock Zeits Für, Verallgemeinerung und L?sung der Diracschen statistischen Gleichung, Phys. 49 (1928) 339
[40] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D35 (1987) 1831
[41] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[42] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A 37 (1988) 2985
[43] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[44] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[45] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J.Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[46] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, (1997)
[47] W. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1974
[48] H. Y. Fan, Entangled states, squeezed states gained via the route of developing dirac’s symbolic method and their applications, Inter. J. Mod. Phys. 18 (2004) 1387
[49] H. P. Yuen, Two—photon coherent states of the radiation field, Phys. Rev. A 13 (1976) 2226
[50] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987)
[51] D. F. Walls, Squeezed states of light, Nature 306 (1983) 141
[52] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 145
[53] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum,Phys. Rev. A 49 (1994) 704
[54] H. Y. Fan and Chen Bo-zhan, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[55] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum, Phys. Rev. A 51 (1995) 3343
[56] H. Y. Fan, H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeezed operators, Phys. Rev. D 35 (1987) 1831
[57] H. Y. Fan and H. Zou, Similarity transformation operators as the images of classical symplectic transformations in coherent state representation, Phys. Lett. A 252 (1999) 281
[1] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[2] M. Orszg, Quantum Optics, Springer-Verlag, Berlin 2000
[3] P. S. Wolfgang, Quantum Optics in Phase Space, Wiley-Vch, Berlin, 2001
[4] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambidge University Press, 1977
[5] G. M. D'Ariano, M. G. Rassetti, J. Katriel, A. I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light (Plenum, New York,), 1989, p.301
[6] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[7] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D 35 (1987) 1831
[8] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[9] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A 37 (1988) 2985
[10] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[11] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[12] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[13] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 145
[14] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum, Phys. Rev. A 49 (1994) 704
[15] H. Y. Fan and B. Z. Chen, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[16] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum,Phys. Rev. A 51 (1995) 3343
[17] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be consideres xomplete?, Phys. Rev. 47 (1935) 777
[18] M. D. Reid, P. D. Drumond. Quantum correlation of phase in nondegenerate parametric oscilation, Phys. Rev. Lett. 69 (1989) 2731
[19] P. D. Drumond, M. D. Reid, Correlation in nondegenerate parametrix oscollation.Ⅱ.below threshold results, Phys. Rev. A 41 (1990) 3930.
[20] Z. Y. Ou, S. F. Pereira, H. J. Kimble and K. C. Peng, Realization of the Einstein-Podolsdy-Rosen for continuous variables, Phys Rev Lett. 68 3663-3666
[21] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[22] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[23] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[24] H. Y. Fan,Squeezed states: Operators for two types of one- and two-mode squeezing transformation, Phys. Rev. A 41 (1990) 1526
[25] H. Y. Fan,Application of entangled state representation in quantum teleportation with continuous variables, Phys. Lett. A 294 (2002) 253
[26] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers 1997
[1] A. Einstein, B. Podolsky and N. Rosen,“Can quantum-mechanical description of physical reality be consideres xomplete?”, Phys. Rev. 47 (1935) 777
[2] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[3] M. Orszg, Quantum Optics, Springer-Verlag, Berlin 2000
[4] P. S. Wolfgang, Quantum Optics in Phase Space, Wiley-Vch, Berlin, 2001
[5] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambidge University Press, 1977
[6] G. M. D’Ariano, M. G. Rassetti, J. Katriel and A. I. Solomon Squeezed and Nonclassical Light ed P. Tombesi and E. R. Pike, Nek York: Plenum, 1989
[7] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[8] H. X. Lu and B. Zhao, Relative entropy as a measure of entanglement for Gaussian states Chin. Phys. 15 (2006) 1914
[9] Y. B. Zhan, Controlled teleportation of high-dimension quantumstate with generalized bell-state measurement, Chin. Phys. 16 (2007) 2557
[10] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootter, Mixed- state entanglement and quant um error correction, Phys. Rev. A 54 (1996) 3824
[11] V. Vedral, M. B. Plenio, M. A. Rippin and P. L..Knight, Quantifying entanglement, Phys. Rev. Lett. 78 (1997) 2275
[12] M. B. Plenio ana V. Vedral, Bounds on relative entropy of entanglement for multi-party systems, J. Phys. A: Math. Gen. 34 (2001) 6997
[13] J. Eisert and H. J. Briegel, Schmidt measure as a tool for quantifying multiparticle entanglement, Phys. Rev. A 64 (2001) 022306
[14] T. C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68 (2003) 042307
[15] L. Tamaryan, D. K. Park, J. W. Son and S. Tamaryan Phys. Rev. A 78 (2008) 032304
[16] H. Y. Fan, A convenient representation for describing an electron in a uniform magnetic field, Phys. Lett. A 126 (1987) 150
[17] H. Y. Fan and J. R. Klauder,Eigenvectors of two particles’relative position and total momentum,Phys. Rev. A 49 (1994) 704
[18] H. Y. Fan and B. Z. Chen, Density matrix for a triatomic linear molecule model,Phys. Rev. A 50 (1994) 3754
[19] H. Y. Fan and X. Ye, Common eigenstates of two particles’center-of-mass coordinates and mass-weighted relative momentum,Phys. Rev. A 51 (1995) 3343
[20] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[21] L. Y. Hu and H. Y. Fan, Two-mode squeezed number state as a two-variable Hermitepolynomial excitation on the squeezed vacuum, J. Mod. Opt. 55 (2008) 2011
[22] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[23] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[24] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[1] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[2] Memorial issue for H. Umezawa, Int. J. Phys. B10 (1996) and reference therein
[3] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[4] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam), 1982
[5] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[6] H. Y. Fan and Y. Fan, New representation of thermal states in thermal field dynamics, Phys. Lett. A246 (1998) 242
[7] H. Y. Fan and Y. Fan, New representation for thermo excitation and de-excitation in thermofield dynamics, Phys. Lett. A 269 (2001) 282
[8] H. Y. Fan, New application of thermo field dynamics in simplifying the calculation of wigner functions, Mod.Phys. Lett. A 18 (2003) 733
[9] H. Y. Fan, New Application of Thermo Field Dynamics in simplifying the calculation of Wigner functions, Mod. Phys. Lett. A, 18 (2003) 733
[10] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[11] H. Y. Fan,H. R. Zaidi and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D35 (1987) 1831
[12] H. Y. Fan and J. R. Klauder, Canonical coherent-state representation of some squeeze operators, J. Phys. A 21 (1988) L725
[13] H. Y. Fan and H. R. Zaidi, Squeezing and frequency jump of a harmonic oscillator, Phys. Rev. A37 (1988) 2985
[14] A. Wünsche, About integration within ordered products in quantum optics,J. Opt, B: Quantum & Semiclass. Opt. 1 (1999) R11
[15] H. Y. Fan and A. Wünsche, Design of squeezing, J. Opt, B: Quantum & Semiclass. Opt. 2 (2000) 464
[16] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J.Opt, B: Quantum & Semiclass. Opt. 7 (2003) R147
[17] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987)
[18] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[19] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[20] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co. 1985
[21] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[22] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[1] E. P. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40 (1932) 749
[2] R. F. O’Connel and E. P. Wigner, Quantum-Mechanical Distribution Functions: Conditions for Uniqueness, Phys. Lett. A 83 (1981) 145
[3] V. Bu?ek, C. H. Keitel. and P. L. Knight, Sampling entropies and operational phase-space measurement. II. Detection of quantum coherences, Phys. Rev. A 51 (1995) 2575
[4] R. P. Feynman Statistical Mechanics (New York:Benjamin) 1972
[5] K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Phys. Rev. A 40 (1989) 2847
[6] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators, Phys. Rev. D 2 (1970) 2161
[7] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space, Phys. Rev. D 2 (1970) 2187
[8] G. S. Agawal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. III. A Generalized Wick Theorem and Multitime Mapping, Phys. Rev. D 2 (1970) 2206
[9] D. F. Walls and G. Miburn, Quantum Optics (Berlin: Springer) 1994
[10] D. T. Smithey, M. Beck, A. Faridani, and M. G. Raymer, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum, Phys. Rev. Lett. 70 (1993) 1244
[11] H. Weyl, Quantenmechanik und Gruppentheorie Z. Phys. 46 (1927) 1-39
[12] M. G. Raymer, Contemporary Physics, 38 (1997)343
[13] K. E. Cahill and R. J. Glauber, Ordered Expansions in Boson Amplitude Operators, Phys. Rev. 177 (1969) 1857
[14] J. R. Klauder B. S. Skagerstam Coherent States. Singapore: World Scientific Publishing Co., 1985
[15] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130 (1963) 2529
[16] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766
[17] H. Y. Fan and H. R. Zaidi, Application of IWOP technique to the generalized Weyl correspondence, Phys. Lett. A 124(1987) 303
[18] H. Y. Fan and T.-N. Ruan, Coherent state formulation of the Weyl correspondence and the Wigner function, Commun. Theor. Phys. (Beijing, China) 2 (1983) 1563
[19] H. Y. Fan and T. N. Ruan, Quantum theory for fermion system and its pseudo-classical correspondence, Commun. Theor. Phys. (Beijing, China) 3 (1984) 45
[20] H. Weyl, Quantum mechanics and group theory, Z. Phys. 46 (1927) 1
[21] H. Y. Fan, Weyl-ordered operator moyal bracket by virtue of method of integral within a Weyl ordered product of operators, Commun. Theor. Phys. (Beijing, China) 3 (2006) 245
[22] H. Y. Fan, Represention and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997
[1] Y. Takahashi and H. Umezawa, Collective Phenomena, 2 (1975) 55
[2] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam), 1982
[3] Memorial issue for H. Umezawa, Int. J. Phys. B10 (1996) and reference therein
[4] H. Umezawa, Advanced Field Theory: Micro, Macro, and Thermal Physics (AIP, New York) 1993
[5] H. Umezawa and Y. Yamanaka, Micro, macro and thermal concepts in quantum field theory, Adv. Phys. 37 (1988) 531
[6] A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Relation Between Quantum and thermal fluctuations, Phys. Lett. A 140 (1989) 475
[7] A. Mann and M. Revzen, Thermal Coherent States, Phys. Lett. A 134 (1989) 273
[8] R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709
[9] G. M. D'Ariano, M. G. Rassetti, J. Katriel, A. I. Solomon, in: P. Tombesi, E.R. Pike (Eds.), Squeezed and Nonclassical Light (Plenum, New York,), 1989, p.301
[10] V. V. Dodonov, Nonclassical states in quantum optics: a squeezed review of the first 75 years (topical review), J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1-R33
[11] H. Y. Fan and Y. Fan, New representation of thermal states in thermal field dynamics, Phys. Lett. A 246 (1998) 242
[12] H. Y. Fan and Fan Yue, New representation for thermo excitation and de-excitation in thermo field dynamics, Phys. Lett. A 269 (2001) 282
[13] H. Y. Fan, New application of thermo field dynamics in simplifying the calculation of wigner functions, Mod.Phys. Lett. A 18 (2003) 733
[14] H. Y. Fan, New Application of Thermo Field Dynamics in simplifying the calculation of Wigner functions, Mod. Phys. Lett. A, 18 (2003) 733
[15] H. Paul, Phase of a microscopic electromagnetic field and its measurement ,Fortschr. Phys. 22, 657 (1974)
[16] H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics Shanghai Scientific & Technical Publishers, 1997