Abstract

We theoretically put forward the Hermite-excited squeezed thermal states (HESTS) by applying operator Hermite polynomials on squeezed thermal states. Starting from the normally ordered density operator of squeezed thermal states and operator Hermite polynomials, the normalization factor is obtained, which is related to the Legendre polynomials. Several phase-space distribution functions, i.e., the Q function, P function, Wigner function (WF), and R function, are analytically derived. And the non-Gaussianity and nonclassicality are mainly reflected by the negativity of non-Gaussian WF and the existence of nonclassical depth. In addition, by deriving the normally ordered density operator and WF of HESTS in the laser channel, the decoherence effect is studied and discussed. Finally, the quantity in measuring non-Gaussianity is calculated to further quantitatively measure the non-Gaussianity of the resulting states.

© 2014 Optical Society of America

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  1. C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
    [CrossRef]
  2. A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, 2005).
  3. F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
    [CrossRef]
  4. M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
    [CrossRef]
  5. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [CrossRef]
  6. S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
    [CrossRef]
  7. S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. 89, 207903 (2002).
    [CrossRef]
  8. R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
    [CrossRef]
  9. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
    [CrossRef]
  10. J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
    [CrossRef]
  11. J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
    [CrossRef]
  12. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
    [CrossRef]
  13. I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
    [CrossRef]
  14. G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
    [CrossRef]
  15. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [CrossRef]
  16. G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
    [CrossRef]
  17. G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
    [CrossRef]
  18. A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
    [CrossRef]
  19. A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
    [CrossRef]
  20. S. M. Barnett and D. T. Pegg, “Phase measurement by projection synthesis,” Phys. Rev. Lett. 76, 4148–4150 (1996).
    [CrossRef]
  21. J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
    [CrossRef]
  22. H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
    [CrossRef]
  23. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
  24. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  25. H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
    [CrossRef]
  26. H. Y. Fan and T. T. Wang, “New operator identities and integration formulas regarding to Hermite polynomials obtained via the operator ordering method,” Int. J. Theor. Phys. 48, 441–448 (2009).
    [CrossRef]
  27. A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1995).
  28. H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
    [CrossRef]
  29. R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).
  30. S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
    [CrossRef]
  31. M. Ban, “Photon statistics of conditional output states of lossless beam splitter,” J. Mod. Opt. 43, 1281–1303 (1996).
    [CrossRef]
  32. M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
    [CrossRef]
  33. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
    [CrossRef]
  34. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
    [CrossRef]
  35. A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
    [CrossRef]
  36. V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
    [CrossRef]
  37. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  38. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
    [CrossRef]
  39. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  40. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2777 (1991).
    [CrossRef]
  41. V. V. Dodonov, “Nonclassical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
    [CrossRef]
  42. C. L. Methta, “Diagonal coherent state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
    [CrossRef]
  43. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
    [CrossRef]
  44. A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B 6, 396–404 (2004).
    [CrossRef]
  45. T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
    [CrossRef]
  46. S. Scheel and D. G. Welsch, “Entanglement generation and degradation by passive optical devices,” Phys. Rev. A 64, 063811 (2001).
    [CrossRef]
  47. D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
    [CrossRef]
  48. S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
    [CrossRef]
  49. C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  50. H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 243–262 (2008).
    [CrossRef]
  51. H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571–5573 (2008).
    [CrossRef]
  52. H. Y. Fan and Y. Fan, “New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials,” Commun. Theor. Phys. 38, 297–300 (2002).
    [CrossRef]
  53. L. Y. Hu and H. Y. Fan, “Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum,” J. Mod. Opt. 55, 2011–2024 (2008).
    [CrossRef]
  54. L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
    [CrossRef]
  55. L. Y. Hu and Z. M. Zhang, “Nonclassicality and decoherence of photon-added squeezed thermal state in thermal environment,” J. Opt. Soc. Am. B 29, 529–537 (2012).
    [CrossRef]
  56. L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
    [CrossRef]
  57. L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
    [CrossRef]
  58. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
    [CrossRef]
  59. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
    [CrossRef]

2012 (2)

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

L. Y. Hu and Z. M. Zhang, “Nonclassicality and decoherence of photon-added squeezed thermal state in thermal environment,” J. Opt. Soc. Am. B 29, 529–537 (2012).
[CrossRef]

2010 (3)

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
[CrossRef]

2009 (4)

H. Y. Fan and T. T. Wang, “New operator identities and integration formulas regarding to Hermite polynomials obtained via the operator ordering method,” Int. J. Theor. Phys. 48, 441–448 (2009).
[CrossRef]

J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

2008 (5)

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 243–262 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571–5573 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum,” J. Mod. Opt. 55, 2011–2024 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
[CrossRef]

2007 (5)

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

2006 (3)

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

2005 (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

2004 (3)

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

2003 (3)

D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
[CrossRef]

S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
[CrossRef]

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

2002 (7)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
[CrossRef]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. 89, 207903 (2002).
[CrossRef]

V. V. Dodonov, “Nonclassical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

H. Y. Fan and Y. Fan, “New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials,” Commun. Theor. Phys. 38, 297–300 (2002).
[CrossRef]

2001 (3)

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

S. Scheel and D. G. Welsch, “Entanglement generation and degradation by passive optical devices,” Phys. Rev. A 64, 063811 (2001).
[CrossRef]

G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[CrossRef]

1998 (1)

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

1997 (1)

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

1996 (2)

M. Ban, “Photon statistics of conditional output states of lossless beam splitter,” J. Mod. Opt. 43, 1281–1303 (1996).
[CrossRef]

S. M. Barnett and D. T. Pegg, “Phase measurement by projection synthesis,” Phys. Rev. Lett. 76, 4148–4150 (1996).
[CrossRef]

1993 (2)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

1991 (3)

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2777 (1991).
[CrossRef]

J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

1984 (1)

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

1967 (1)

C. L. Methta, “Diagonal coherent state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

1963 (1)

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Anhut, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

Ban, M.

M. Ban, “Photon statistics of conditional output states of lossless beam splitter,” J. Mod. Opt. 43, 1281–1303 (1996).
[CrossRef]

Banaszek, K.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Barbosa, G. A.

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

Barnett, S. M.

S. M. Barnett and D. T. Pegg, “Phase measurement by projection synthesis,” Phys. Rev. Lett. 76, 4148–4150 (1996).
[CrossRef]

Bartlett, S. D.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. 89, 207903 (2002).
[CrossRef]

Bellini, M.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Bergou, J. A.

J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

Björk, G.

G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[CrossRef]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Chuang, I. L.

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Cirac, J. I.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Corndorf, E.

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Dakna, M.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

Dantan, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

De Siena, S.

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Dell’Anno, F.

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, “Nonclassical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

Eisert, J.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

Fan, H. Y.

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

H. Y. Fan and T. T. Wang, “New operator identities and integration formulas regarding to Hermite polynomials obtained via the operator ordering method,” Int. J. Theor. Phys. 48, 441–448 (2009).
[CrossRef]

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum,” J. Mod. Opt. 55, 2011–2024 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571–5573 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 243–262 (2008).
[CrossRef]

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

H. Y. Fan and Y. Fan, “New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials,” Commun. Theor. Phys. 38, 297–300 (2002).
[CrossRef]

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

Fan, Y.

H. Y. Fan and Y. Fan, “New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials,” Commun. Theor. Phys. 38, 297–300 (2002).
[CrossRef]

Ferraro, A.

A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, 2005).

Fiurášek, J.

J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
[CrossRef]

Furusawa, A.

Garcia-Patron, R.

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Garci-Patrón, R.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

Gardiner, C.

C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Genoni, M. G.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Giedke, G.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Grangier, P.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Hillery, M.

J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Hiroshima, T.

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

Hu, L. Y.

L. Y. Hu and Z. M. Zhang, “Nonclassicality and decoherence of photon-added squeezed thermal state in thermal environment,” J. Opt. Soc. Am. B 29, 529–537 (2012).
[CrossRef]

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum,” J. Mod. Opt. 55, 2011–2024 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571–5573 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 243–262 (2008).
[CrossRef]

Illuminati, F.

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Jozsa, R.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Kenfack, A.

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

Kim, M. S.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
[CrossRef]

Knoll, L.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

Kumar, P.

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

Lee, C. T.

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2777 (1991).
[CrossRef]

Lee, J.

D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
[CrossRef]

Lee, S. Y.

S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
[CrossRef]

Leung, D. W.

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Meng, X. G.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

Methta, C. L.

C. L. Methta, “Diagonal coherent state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

Nemoto, K.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

Nha, H.

S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
[CrossRef]

Niset, J.

J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Olivares, S.

S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
[CrossRef]

A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, 2005).

Opatrny, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

Parigi, V.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

Paris, M. G. A.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
[CrossRef]

A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, 2005).

Pegg, D. T.

S. M. Barnett and D. T. Pegg, “Phase measurement by projection synthesis,” Phys. Rev. Lett. 76, 4148–4150 (1996).
[CrossRef]

Peres, A.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1995).

Pirandola, S.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

Plenio, M. B.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Rossi, A. R.

S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
[CrossRef]

Sáchez-Soto, L. L.

G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[CrossRef]

Sanders, B. C.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. 89, 207903 (2002).
[CrossRef]

Sasaki, M.

Scheel, S.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

S. Scheel and D. G. Welsch, “Entanglement generation and degradation by passive optical devices,” Phys. Rev. A 64, 063811 (2001).
[CrossRef]

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

Söerholm, J.

G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[CrossRef]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[CrossRef]

Takahashi, H.

Tara, K.

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Tualle-Brouri, R.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

van Loock, P.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

Vandersypen, L. M. K.

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Wakui, K.

Wang, J. S.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

Wang, T. T.

H. Y. Fan and T. T. Wang, “New operator identities and integration formulas regarding to Hermite polynomials obtained via the operator ordering method,” Int. J. Theor. Phys. 48, 441–448 (2009).
[CrossRef]

Wang, Z. S.

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Weedbrook, C.

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

Welsch, D. G.

S. Scheel and D. G. Welsch, “Entanglement generation and degradation by passive optical devices,” Phys. Rev. A 64, 063811 (2001).
[CrossRef]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

Wenger, J.

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Wilson, D.

D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
[CrossRef]

Wolf, M. M.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Wootters, W. K.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Xu, X. F.

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Xu, X. X.

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Ye, X.

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

Yu, D.

J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

Yuen, H. P.

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

Zavatta, A.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Zhang, Z. M.

Zhou, X.

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Zoller, P.

C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

Zyczkowski, K.

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

Ann. Phys. (1)

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
[CrossRef]

Commun. Theor. Phys. (2)

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

H. Y. Fan and Y. Fan, “New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials,” Commun. Theor. Phys. 38, 297–300 (2002).
[CrossRef]

Int. J. Theor. Phys. (1)

H. Y. Fan and T. T. Wang, “New operator identities and integration formulas regarding to Hermite polynomials obtained via the operator ordering method,” Int. J. Theor. Phys. 48, 441–448 (2009).
[CrossRef]

J. Mod. Opt. (4)

M. Ban, “Photon statistics of conditional output states of lossless beam splitter,” J. Mod. Opt. 43, 1281–1303 (1996).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum,” J. Mod. Opt. 55, 2011–2024 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

D. Wilson, J. Lee, and M. S. Kim, “Entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment,” J. Mod. Opt. 50, 1809–1815 (2003).
[CrossRef]

J. Opt. B (2)

V. V. Dodonov, “Nonclassical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of non-classicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Mod. Phys. Lett. B (1)

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 243–262 (2008).
[CrossRef]

Nature (1)

I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393, 143–146 (1998).
[CrossRef]

Opt. Commun. (2)

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571–5573 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (2)

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

S. Olivares, M. G. A. Paris, and A. R. Rossi, “Optimized teleportation in Gaussian noisy channels,” Phys. Lett. A 319, 32–43 (2003).
[CrossRef]

Phys. Rep. (2)

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (12)

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2777 (1991).
[CrossRef]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D. G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184–3194 (1997).
[CrossRef]

S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82, 053812 (2010).
[CrossRef]

J. A. Bergou, M. Hillery, and D. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

S. Scheel and D. G. Welsch, “Entanglement generation and degradation by passive optical devices,” Phys. Rev. A 64, 063811 (2001).
[CrossRef]

L. Y. Hu, X. X. Xu, Z. S. Wang, and X. F. Xu, “Photon-added squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

Phys. Rev. Lett. (14)

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[CrossRef]

C. L. Methta, “Diagonal coherent state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

S. M. Barnett and D. T. Pegg, “Phase measurement by projection synthesis,” Phys. Rev. Lett. 76, 4148–4150 (1996).
[CrossRef]

G. Björk, L. L. Sáchez-Soto, and J. Söerholm, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[CrossRef]

G. A. Barbosa, E. Corndorf, P. Kumar, and H. P. Yuen, “Secure communication using mesoscopic coherent states,” Phys. Rev. Lett. 90, 227901 (2003).
[CrossRef]

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Efficient classical simulation of optical quantum information circuits,” Phys. Rev. Lett. 89, 207903 (2002).
[CrossRef]

R. Garcia-Patron, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
[CrossRef]

J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009).
[CrossRef]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

Rev. Mod. Phys. (4)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

C. Weedbrook, S. Pirandola, R. Garci-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Science (2)

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Other (6)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1995).

A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, 2005).

C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

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Figures (9)

Fig. 1.
Fig. 1.

Q function of HESTS for different values of m: (a) m=1, (b) m=2, (c) m=3, and (d) m=4 with ϵ=0.4, r=0.3, and n¯=0.3, respectively.

Fig. 2.
Fig. 2.

WF of HESTS for different values of m: (a) m=1, (b) m=2, (c) m=3, and (d) m=4, with ϵ=0.4, r=0.3, and n¯=0.3, respectively.

Fig. 3.
Fig. 3.

WF of HESTS for different values of m, r, and n¯: (a) m=1, r=0.5, and n¯=0.3; (b) m=1, r=0.3, and n¯=0.8; (c) m=2, r=0.5, and n¯=0.3; and (d) m=2, r=0.3, and n¯=0.8, with ϵ=0.4, respectively.

Fig. 4.
Fig. 4.

R function of HESTS for different values of m and τ: (a) m=1 and τ=0.3, (b) m=1 and τ=0.9, (c) m=2 and τ=0.3, and (d) m=2 and τ=0.9 with ϵ=0.4, r=0.3, and n¯=0.3, respectively.

Fig. 5.
Fig. 5.

Evolution of negative volume for R function versus τ for m=1, 2, and 3 (corresponding to lower to upper curves marked with an arrow, respectively) with ϵ=0.4, r=0.3, and n¯=0.3.

Fig. 6.
Fig. 6.

WF of HESTS in laser channel for different values of t, (a) t=0.03, (b) t=0.14 with k=1 and g=2, and that for different values of k and g, (c) k=6 and g=2 and (d) k=1 and g=10 with t=0.03, with m=1, ϵ=0.4, r=0.3, and n¯=0.3, respectively.

Fig. 7.
Fig. 7.

Evolution of negative volume for WF versus t for m=1, 2, 3, and 4 (corresponding to lower to upper curves marked with an arrow, respectively) with ϵ=0.4, r=0.3, and n¯=0.3, k=1, g=2.

Fig. 8.
Fig. 8.

Evolution of threshold value of decoherence time scale versus r for HESTS and PSSTS in thermal channel, respectively, with m=1, n¯=0.3, and nc=0.5.

Fig. 9.
Fig. 9.

Quantity in measuring non-Gaussianity versus (a) r with ϵ=0.4 and n¯=0.3 and (b) n¯ with ϵ=0.4 and r=0.3 for several different values of m: m=1, 2, 3, and 4 (corresponding to upper to lower curves marked with an arrow, respectively).

Equations (53)

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ρm=(Nm)1Hm(ϵQ)ρsHm(ϵQ),
ρs=S(r)ρthS(r),
ρth=(1eσ)eσaa,
eσaa=:exp[(eσ1)aa]:
eσaa=2eσ+1::exp[eσ1eσ+1(Q2+P2)]::,
ρs=1τ1τ2:exp(Q22τ12P22τ22):,
exp(t2+2xt)=n=0tnn!Hn(x)
eAeB=eA+Be12[A,B],[A,[A,B]]=[B,[A,B]]=0,
m=0tmm!Hm(ϵQ)=m=0(t1ϵ2)mm!:Hm(ϵQ1ϵ2):,ϵ1.
Hm(ϵQ)=(1ϵ2)m2:Hm(ϵQ1ϵ2):.
Pm(x)=m!l=0[m2]xm22l(l!)2(m2l)!(11x2)l,
d2zπexp(ζ|z|2+ξz+ηz*+fz2+gz*2)=1ζ24fgexp(ζξη+ξ2g+η2fζ24fg),
Nm=m![B(1ϵ2)]mPm(A/B),
A=2ϵ21ϵ2(2τ121),B=2ϵ21ϵ2(τ121)1.
H1(ϵQ)=2ϵ(a+a),H2(ϵQ)=2ϵ2(a2+aa+aa+a2)2.
Q(α,α*)=1πα|ρ|α.
Hm(x)=mtmexp(2xtt2)|t=0,
Q(α,α*)=Qm(α,α*)Q0(α,α*),
Qm(α,α*)=(Nm)1(m!)2[A1(1ϵ2)]ml=0m(A0/A1)ll![(ml)!]2|Hml(iA22A1)|2
Q0(α,α*)=1π1τ1τ2exp[A3|α|2+A4(α2+α*2)]
A0=2ϵ21ϵ2(1+A3),A1=2ϵ2(1ϵ2)A41,A2=2ϵ2(1ϵ2)[(1+A3)α+(12A4)α*],A3=τ12+τ222τ12τ22,A4=τ12τ224τ12τ22.
A^=d2zπz|A^|zexp(|z|2+z*aza+aa),
ρm=X0τ1τ2(m!)2Nm[B1(1ϵ2)]ml=0m(B0/B1)ll![(ml)!]2|Hml(iB22B1)|2exp[B3aa+B4(a2+a2)],
P(α,α*)=X0τ1τ2(m!)2Nm[B1(1ϵ2)]ml=0m(B0/B1)ll![(ml)!]2|Hml(iB22B1)|2exp[B3|α|2+B4(α2+α*2)],
B0=2ϵ21ϵ22τ12τ22τ12τ222(τ121)(τ221),B1=2ϵ21ϵ2(B41)1,B2=2ϵ1ϵ2[(1+2B4)α+(1B3)α*],B3=2τ12τ222(τ121)(τ221),B4=τ12τ224(τ121)(τ221),X0=τ12τ22(τ121)(τ221).
W(α,α*)=Tr[ρΔ(α,α*)],
Δ(α,α*)=d2zπ2|α+zαz|eαz*α*z.
W(α,α*)=Fm(α,α*)W0(α,α*),
W0(α,α*)=1π(2n¯+1)exp[2cosh2r2n¯+1|α|2+sinh2r2n¯+1(α2+α*2)]
Fm(α,α*)=(Nm)1(m!)2[C1(1ϵ2)]m×l=0m(C0/C1)ll![(ml)!]2|Hml(iC22C1)|2
C0=2ϵ21ϵ212τ221,C1=ϵ21ϵ2(12τ2211)1,C2=4ϵ2(1ϵ2)12τ221[(τ221)α+τ22α*].
R(α,τ)=1τd2zπexp(1τ|αz|2)P(z,z*).
R(α,τ)=X0X1ττ1τ2(Nm)1(m!)2[D1(1ϵ2)]ml=0m(D0/D1)ll![(ml)!]2|Hml(iD22D1)|2exp[D3|α|2+D4(α2+α*2)],
D0=2ϵ21ϵ2X02X1[g1(g0g1+4h1X0A4)+g0h12]+B0,D1=2ϵ21ϵ2X02X1[X0A4(g1+h12)+g0g1h1]+B1,D2=2ϵX0X1τ1ϵ2[(g0g1+2h1X0A4)α+(g0h1+2g1X0A4)α*],D3=1τ(1τg0X11),D4=1τ2X0X1A4,g0=1τ+h1X01,g1=h122A4(2A4+1),h1=1+A3,X1=[g024(X0A4)2]1.
δ=12[d2α|R(α,α*)|1].
dρ(t)dt=g[2aρ(t)aaaρ(t)ρ(t)aa]+k[2aρ(t)aaaρ(t)ρ(t)aa],
ρ(t)=T3j,l=0(gT1)j(kT1)lj!l!ajexp(aalnT2)×alρ0alexp(aalnT2)aj,
T1=1e2(kg)tkge2(kg)t,T2=(kg)e(kg)tkge2(kg)t,T3=kgkge2(kg)t,
ρ(t)=j,l=0Mj,lρ0Mj,l
Mj,lT3(gT1)j(kT1)lj!l!ajexp(aalnT2)al.
j,lMj,lMj,l=exp(2aalnT2)exp(2aalnT2)=1.
W(α,α*;t)=2Ad2zπW(z,z*;0)exp[2A|αze(kg)t|2],
W(α,α*;t)=Fm(α,α*;t)W0(α,α*;t),
Fm(α,α*;t)=(Nm)1(m!)2[A˜1(1ϵ2)]ml=0m(A˜0/A˜1)ll![(ml)!]2|Hml(iA˜22A˜1)|2
W0(α,α*;t)=1π2XA(2n¯+1)exp[A˜3|α|2+A˜4(α2+α*2)]
A˜0=8ϵ2X1ϵ21(2τ221)2[k˜0k˜1+4sinh2r2n¯+1τ22(τ221)]+C0,A˜1=8ϵ2X1ϵ21(2τ221)2[k˜0τ22(τ221)+sinh2r2n¯+1k˜1]C1,A˜2=8ϵXe(kg)tA2(1ϵ2)12τ221{[k˜0(τ221)+2sinh2r2n¯+1τ22]α+[k˜0τ22+2sinh2r2n¯+1(τ221)]α*},A˜3=2A[2Ak˜0Xe2(kg)t1],A˜4=4Xsinh2rA2(2n¯+1)e2(kg)t,k˜0=2[1Ae2(kg)t+cosh2r2n¯+1],k˜1=(τ221)2+(τ22)2,X={4[1Ae2(kg)t+e2r2n¯+1][1Ae2(kg)t+e2r2n¯+1]}1.
tc=12(kg)ln[1+kgk+gX˜(2n¯+1)],
κtch=12ln(1+2n¯+12nc+1X˜),
κtcs=12ln(12n¯+12nc+1n¯sinh2rn¯cosh2r+sinh2r).
F=Tr(ρρm)/Tr(ρ2).
Tr(ρρm)=4πd2αW(α,α*)W0(α,α*),Tr(ρ2)=4πd2α[W0(α,α*)]2,
F=(Nm)1m![B˜1(1ϵ2)]mPm(B˜0/B˜1),
B˜0=2ϵ2e2r(1ϵ2)(2n¯+1)[k˜1e2rcosh2r+2τ22(τ221)e2rsinh2r1],B˜1=ϵ2e2r(1ϵ2)(2n¯+1)[k˜1e2rsinh2r+1+2τ22(τ221)e2rcosh2r]11ϵ2.

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