Abstract

We introduce a kind of entangled state—a photon-addition two-mode squeezed thermal state (TMSTS)—by adding photons to each mode of the TMSTS. Using the P-representation of thermal state, the compact expression of the normalization factor is derived, a Jacobi polynomial. The nonclassicality is investigated by exploring especially the negativity of Wigner function. The entanglement is discussed by using Shchukin–Vogel criteria. It is shown that the photon addition to the TMSTS may be more effective for the entanglement enhancement than the photon subtraction from the TMSTS. In addition, the quantum teleportation is also examined, which shows that symmetrical photon-added TMSTS may be more useful for quantum teleportation than the nonsymmetric case.

© 2012 Optical Society of America

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  3. G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
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  4. J. Fiurasek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
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  5. T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
    [CrossRef]
  6. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-toclassical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
    [CrossRef]
  7. 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, 023820–023828 (2005).
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  8. H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
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  9. J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
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  10. M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
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  11. L. Y. Hu and H. Y. Fan, “Statistical properties of photonsubtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
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  12. 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]
  13. S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
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  14. A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
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  15. A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502–030505(2007).
    [CrossRef]
  16. A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
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  17. L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
    [CrossRef]
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  23. 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).
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  28. H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
    [CrossRef]
  29. F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
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    [CrossRef]
  32. P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
    [CrossRef]
  33. P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
    [CrossRef]
  34. Z. X. Zhang and H. Y. Fan, “Some properties of states engendered by the excitations on a two-mode squeezed vacuum state,” Phys. Lett. A 174, 206–209 (1993).
    [CrossRef]
  35. W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
    [CrossRef]
  36. M. G. Benedict and A. Czirjak, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A 60, 4034–4044 (1999).
    [CrossRef]
  37. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
    [CrossRef]
  38. J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
    [CrossRef]
  39. E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
    [CrossRef]
  40. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
    [CrossRef]
  41. C. T. Lee, “Many-photon anti-bunching in generalized pair coherent states,” Phys. Rev. A 41, 1569–1575 (1990).
    [CrossRef]
  42. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  43. D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
    [CrossRef]
  44. R. García-Patrón, 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]
  45. X. Y. Chen, “The entanglement properties of non-Gaussian states prepared by photon subtraction from two-mode squeezed thermal states,” Phys. Lett. A 372, 2976–2979 (2008).
    [CrossRef]
  46. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [CrossRef]
  47. P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A 74, 042306 (2006).
    [CrossRef]
  48. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
    [CrossRef]
  49. Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
    [CrossRef]

2012

2011

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

2010

2009

Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

2008

H. Y. Fan and L. Y. Hu, “Two quantum-mechanical photocount formulas,” Opt. Lett. 33, 443–445 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Statistical properties of photonsubtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

X. Y. Chen, “The entanglement properties of non-Gaussian states prepared by photon subtraction from two-mode squeezed thermal states,” Phys. Lett. A 372, 2976–2979 (2008).
[CrossRef]

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[CrossRef]

2007

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. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502–030505(2007).
[CrossRef]

2006

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807–043824 (2006).
[CrossRef]

P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A 74, 042306 (2006).
[CrossRef]

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

2005

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

J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef]

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (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, 023820–023828 (2005).
[CrossRef]

2004

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

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

R. García-Patrón, 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]

2003

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
[CrossRef]

2002

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

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

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

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements onthe two-mode squeezed vacuum,” Phys. Rev. A 65, 062306–062311 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Universal continuous variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304–042308(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]

2001

P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
[CrossRef]

2000

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

1999

M. G. Benedict and A. Czirjak, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A 60, 4034–4044 (1999).
[CrossRef]

1998

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

1996

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
[CrossRef]

1993

Z. X. Zhang and H. Y. Fan, “Some properties of states engendered by the excitations on a two-mode squeezed vacuum state,” Phys. Lett. A 174, 206–209 (1993).
[CrossRef]

1992

P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
[CrossRef]

1991

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (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]

1990

W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

C. T. Lee, “Many-photon anti-bunching in generalized pair coherent states,” Phys. Rev. A 41, 1569–1575 (1990).
[CrossRef]

1987

F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

1932

E. P. 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]

Asboth, J. K.

J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef]

Banaszek, K.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (2005).
[CrossRef]

Barnett, S. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

Bartlett, S. D.

S. D. Bartlett and B. C. Sanders, “Universal continuous variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304–042308(2002).
[CrossRef]

Bellini, M.

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, 023820–023828 (2005).
[CrossRef]

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

Benedict, M. G.

M. G. Benedict and A. Czirjak, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A 60, 4034–4044 (1999).
[CrossRef]

Bonifacio, R.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
[CrossRef]

Bouwmeester, D.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Braunstein, S. L.

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

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

Browne, D. E.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Calsamiglia, J.

J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef]

Carmichael, H. J.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

Cerf, N. J.

R. García-Patrón, 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]

Chefles, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

Chen, X. Y.

X. Y. Chen, “The entanglement properties of non-Gaussian states prepared by photon subtraction from two-mode squeezed thermal states,” Phys. Lett. A 372, 2976–2979 (2008).
[CrossRef]

Cirac, J. I.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

Cochrane, P. T.

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements onthe two-mode squeezed vacuum,” Phys. Rev. A 65, 062306–062311 (2002).
[CrossRef]

Czirjak, A.

M. G. Benedict and A. Czirjak, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A 60, 4034–4044 (1999).
[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, 030502–030505(2007).
[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.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

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

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Fan, H. Y.

L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Two quantum-mechanical photocount formulas,” Opt. Lett. 33, 443–445 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Statistical properties of photonsubtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

Z. X. Zhang and H. Y. Fan, “Some properties of states engendered by the excitations on a two-mode squeezed vacuum state,” Phys. Lett. A 174, 206–209 (1993).
[CrossRef]

Fan, Y.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

Feng, D. F.

W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Fiurasek, J.

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

Fiurášek, J.

R. García-Patrón, 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]

García-Patrón, R.

R. García-Patrón, 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]

Giedke, G.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

Gilmore, R.

W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[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, 030502–030505(2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

R. García-Patrón, 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]

Hong-Yi, F.

F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Hu, L. Y.

Invernizzi, C.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (2005).
[CrossRef]

Ji, S. W.

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

Kim, H. J.

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

Kim, M. S.

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[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]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

Kitagawa, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

Klauder, J. R.

F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Klyshko, D. N.

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
[CrossRef]

Kurizki, G.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Lee, C. T.

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
[CrossRef]

C. T. Lee, “Many-photon anti-bunching in generalized pair coherent states,” Phys. Rev. A 41, 1569–1575 (1990).
[CrossRef]

Lee, S. Y.

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

Li, F. L.

Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

Lu, H. L.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

Marian, P.

P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A 74, 042306 (2006).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
[CrossRef]

P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
[CrossRef]

Marian, T. A.

P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A 74, 042306 (2006).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
[CrossRef]

Milburn, G. J.

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements onthe two-mode squeezed vacuum,” Phys. Rev. A 65, 062306–062311 (2002).
[CrossRef]

Nha, H.

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

Olivares, S.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (2005).
[CrossRef]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
[CrossRef]

Opatrný, T.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[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, 030502–030505(2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
[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]

Paris, M. G. A.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (2005).
[CrossRef]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
[CrossRef]

Plenio, M. B.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

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), Appendix A.

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

Ralph, T. C.

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements onthe two-mode squeezed vacuum,” Phys. Rev. A 65, 062306–062311 (2002).
[CrossRef]

Ritsch, H.

J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef]

Sanders, B. C.

S. D. Bartlett and B. C. Sanders, “Universal continuous variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304–042308(2002).
[CrossRef]

Sasaki, M.

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807–043824 (2006).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

Scheel, S.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

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

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

Scutaru, H.

P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
[CrossRef]

Shchukin, E.

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

Suzuki, S.

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807–043824 (2006).
[CrossRef]

Takeoka, M.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

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]

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, 030502–030505(2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

R. García-Patrón, 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]

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, 023820–023828 (2005).
[CrossRef]

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

Vogel, W.

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

Welsch, D.-G.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

R. García-Patrón, 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. P.

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

Xu, X. X.

Yang, Y.

Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

Zaidi, H. R.

F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Zavatta, A.

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, 023820–023828 (2005).
[CrossRef]

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

Zeilinger, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Zhang, W. M.

W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Zhang, Z. M.

Zhang, Z. X.

Z. X. Zhang and H. Y. Fan, “Some properties of states engendered by the excitations on a two-mode squeezed vacuum state,” Phys. Lett. A 174, 206–209 (1993).
[CrossRef]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

Ann. Phys.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

J. Opt. B

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

J. Opt. Soc. Am. B

J. Phys. A: Math. Gen.

P. Marian, T. A. Marian, and H. Scutaru, “Inseparability of mixed two-mode Gaussian states generated with a SU(1, 1) interferometer,” J. Phys. A: Math. Gen. 34, 6969–6980 (2001).
[CrossRef]

J. Phys. B

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[CrossRef]

Opt. Lett.

Phys. Lett. A

Z. X. Zhang and H. Y. Fan, “Some properties of states engendered by the excitations on a two-mode squeezed vacuum state,” Phys. Lett. A 174, 206–209 (1993).
[CrossRef]

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
[CrossRef]

X. Y. Chen, “The entanglement properties of non-Gaussian states prepared by photon subtraction from two-mode squeezed thermal states,” Phys. Lett. A 372, 2976–2979 (2008).
[CrossRef]

Phys. Rev.

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

Phys. Rev. A

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

C. T. Lee, “Many-photon anti-bunching in generalized pair coherent states,” Phys. Rev. A 41, 1569–1575 (1990).
[CrossRef]

P. Marian and T. A. Marian, “Continuous-variable teleportation in the characteristic-function description,” Phys. Rev. A 74, 042306 (2006).
[CrossRef]

M. G. Benedict and A. Czirjak, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A 60, 4034–4044 (1999).
[CrossRef]

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
[CrossRef]

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[CrossRef]

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[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, 023820–023828 (2005).
[CrossRef]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314–032318 (2003).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310–042321 (2006).
[CrossRef]

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements onthe two-mode squeezed vacuum,” Phys. Rev. A 65, 062306–062311 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Universal continuous variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304–042308(2002).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807–043824 (2006).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105–042116 (2005).
[CrossRef]

S. Y. Lee, S. W. Ji, H. J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84, 012302 (2011).
[CrossRef]

P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
[CrossRef]

Y. Yang and F. L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[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]

Phys. Rev. D

F. Hong-Yi, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Phys. Rev. Lett.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502–030505(2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601–213604 (2006).
[CrossRef]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

J. Fiurasek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (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. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef]

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

R. García-Patrón, 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]

Rev. Mod. Phys.

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

W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Science

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-toclassical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Other

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

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

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

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Figures (6)

Fig. 1.
Fig. 1.

Cross-correlation function between the two modes a and b as a function of r for different parameters (m,n) and n¯=0.01.

Fig. 2.
Fig. 2.

Rab as a function of r for different parameters (m,n) and n¯=0.01.

Fig. 3.
Fig. 3.

The WF W(α,β) in phase space (Q+,P+) for several different parameter values (m,n) and n¯ with r=0.3. (a) m=0, n=1, n¯=0.2; (b) m=0, n=1, n¯=1; (c) m=n=1, n¯=0.2; and (d) m=n=1, n¯=1.

Fig. 4.
Fig. 4.

The WF W(α,β) in phase space (Q+,P+) for several different parameter values (m,n) with n¯=0.2 and r=0.3. (a) m=0, n=2; (b) m=1, n=2; (c) m=1, n=3; and (d) m=2, n=3.

Fig. 5.
Fig. 5.

The sufficient condition of inseparability as the function of r and n¯ for (a) m=0 and n=1; (b) m=n=1.

Fig. 6.
Fig. 6.

The fidelity as the function of r for several different (m,n) values and n¯=0.01.

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

ρSANm,n1ambnS(r)ρth1ρth2S(r)ambn,
ρth1,2=n=0n¯n(n¯+1)n+1|nn|,
ρthj=1n¯d2απe1n¯|α|2|αα|,
S(r)aS(r)=acoshrbsinhr,S(r)bS(r)=bcoshrasinhr.
S|α,β=sechrexp[12(|α|2+|β|2)]×exp[α(acoshrbsinhr)]×exp[β(bcoshrasinhr)]×exp[abtanhr]|00,
S|α,β=sechrexp[12(|α|2+|β|2)αβtanhr]×exp[(aα+bβ)sechr+abtanhr]|00.
ρSSρth1ρth2S=A1exp[A2(ab+ab)A3(aa+bb)],
A1=sech2r(n¯+1)2n¯2tanh2r,A2=(2n¯+1)sinhrcoshr(2n¯+1)cosh2r+n¯2,A3=cosh2r+n¯cosh2r(2n¯+1)cosh2r+n¯2,
d2zπeζ|z|2+ξz+ηz*=1ζeξηζ,Reζ<0.
O^=d2zπz|O^|ze|z|2+z*aza+aa,
ρS=A˜exp[A˜2(ab+ab)A˜3(aa+bb)],
A˜1=1(n¯+1)2(2n¯+1)cosh2r,A˜2=(2n¯+1)sinhrcoshr(n¯+1)2(2n¯+1)cosh2r,A˜3=sinh2r+n¯cosh2r(n¯+1)2(2n¯+1)cosh2r.
P(α,β)=A1˜exp[A2˜(α*β*+αβ)A3˜(|α|2+|β|2)],
ρS=d2αd2βπ2P(α,β)|α,βα,β|.
ρS(r=0)=1(n¯+1)2exp[aa+bbn¯+1]=1n¯2exp[1n¯(aa+bb)],
ρS(n¯=0)=sech2rexp[(ab+ab)tanhr(aa+bb)]=csch2rexp[aa+bb(ab+ab)cothr],
ρSA=A1Nm,nambneA2(ab+ab)A3(aa+bb)ambn.
Nm,n=2m+2nτmtmτntneB1(τt+τt)+B2(ττ+tt)|t,τ,t,τ=0,
B1=cosh2r+n¯cosh2r,B2=(2n¯+1)sinhrcoshr.
2m+2nτmtmτntneA(τt+τt)+B(ττ+tt)|t,τ,t,τ=0=m!n!{Anm(B2A2)mPm(nm,0)(B2+A2B2A2)mn,Amn(B2A2)nPn(mn,0)(B2+A2B2A2)nm;
Nm,n=m!n!B1nmωmPm(0,nm)(υω),
ω=n¯2+(2n¯+1)cosh2r,υ=n¯+(n¯+1)cosh4r+(n¯+cosh2r)cosh2r.
Nm,n(n¯=0)=m!n!cosh2nrPm(0,nm)(cosh2r),
aa=Nm+1,nNm,n1,bb=Nm,n+1Nm,n1,
abab=Nm+1,n+1Nm+1,nNm,n+1Nm,n+1.
gm,n(r)=Nm+1,n+1Nm,nNm,n+1Nm+1,n(Nm,n+1Nm,n)(Nm+1,nNm,n).
P(ma,nb)=A1[n¯(n¯+1)]nbmaμmaνnaPma(nbma,0)(χ),
ν=(2n¯+1)cosh2r+n¯2,μ=(2n¯+1)cosh2r(n¯+1)2,χ=((2n¯+1)sinh2r)2+4n¯2(n¯+1)2((2n¯+1)sinh2r)24n¯2(n¯+1)2.
Pn¯0(ma,nb)=tanh2marcosh2rδma,na,
Pr0(ma,nb)=n¯nb(n¯+1)nb+1n¯ma(n¯+1)ma+1,
P¯SA(ma,nb)=Nm,n1ma!nb!(mam)!(nbn)!P(mam,nbn).
Raba2a2+b2b22aabb1<0.
a2a2=Nm+2,nNm,n4Nm+1,nNm,n+2,b2b2=Nm,n+2Nm,n4Nm,n+1Nm,n+2.
Rab=Nm+2,n+Nm,n+2+2(ΩNm+1,n+1)2(Nm+1,n+1+Ω),(Ω=Nm,nNm+1,nNm,n+1).
Rab,m=n=0=(2n¯+1)(4cosh2r+(2n¯+1)sinh22r(2n¯+1)[(2n¯+1)cosh4r2cosh2r]+1.
W(α,β)=e2(|α|2+|β|2)d2z1d2z2π4z1,z2|ρ|z1,z2×exp[2(αz1*α*z1)+2(βz2*β*z2)],
Wm,n(α,β)=W0(α,β)Fm,n(α,β),
W0(α,β)=π2(2n¯+1)2exp{2cosh2r2n¯+1(|α|2+|β|2)+2sinh2r2n¯+1(βα+α*β*)},
Fm,n(α,β)=K3m+nNm,nl=0mj=0n(m!n!)2(K1/K3)l+jl!j![(ml)!(nj)!]2×|Hml,nj(R1iK3,R3iK3)|2,
R1=2(K1αK3β*),R3=2(K1βK3α*),K1=n¯+cosh2r2n¯+1,K3=sinhrcoshr2n¯+1.
W0,n(α,β)=W0(α,β)(K1/B1)nLn(|R3|2/K1),
W0,1(0,0)=(n¯+cosh2r)/(2n¯+1)3(cosh2r+n¯cosh2r)π2,
SVm,naa12bb12abab<0.
ab=Nm,m+1,n,n+1Nm,n,ab=Nm+1,m,n+1,nNm,n,
Nl,p,q,rl+p+q+rτltpτqtre(τt+τt)B1+(ττ+tt)B2|t,τ,t,τ=0=s=0l!r!q!p!B1lq+2rB2qr(B22/B12)sδp+q,l+rs!(qr+s)!(rs)!(l+rqs)!.
SVm,n=(Nm+1,nNm,n32)(Nm,n+1Nm,n32)Nm,m+1,n,n+1Nm,nNm+1,m,n+1,nNm,n.
SV0,1=(υB132)(2B132)4B22.
SV1,1=(B1(3ωυ)32)24(2B12+B22)2B22υ2.
χE(α,β)=1Nm,ne(B112)(|α|2+|β|2)+B2(αβ+α*β*)×2m+2nτmτntmtneB1(τt+tτ)+B2(tt+ττ)×et(αB1B2β*)+τ(βB2B1α*)×eτ(αB2B1β*)+t(βB1B2α*)|t,τ,t,τ=0,
χout(η)=χin(η)χE(η*,η),
F=d2ηπχin(η)χout(η).
Fm,nn¯=[(2n¯+1)e2r+1]m+n(2n¯+1)e2r+1(m+n)!22m+2nNm,n.
F0,0n¯=1/[(2n¯+1)e2r+1],
Fm,n0=(e2r+1)n+me2r+1(m+n)!22m+2nNm,n.
F0,00=(1+tanhr)/2,F1,10=(1+tanhr)34(1+tanh2r),F0,10=(1+tanhr)4(1+tanhr)sech2r.

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