Abstract

We propose a specific method for converting a four-photon Greenberger-Horne-Zeilinger (GHZ) state to a W state in a deterministic way by using linear optical elements, cross-Kerr nonlinearities, and homodyne measurement. We consider the effects of the quadrature homodyne measurements on the fidelity of the W state and the experimental feasibility of the proposed scheme. This might provide great prospects for converting multipartite entangled states into each other for future optical quantum information processing (QIP).

© 2016 Optical Society of America

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    [Crossref]
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  6. H. Q. Ma, K. J. Wei, and J. H. Yang, “Experimental circular quantum secret sharing over telecom fiber network,” Opt. Express 21, 16663–16669 (2013).
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  7. Q. Y. Cai and B. W. Li, “Improving the capacity of the Boström-Felbinger protocol,” Phys. Rev. A 69, 054301 (2004).
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  8. A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
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  9. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
    [Crossref]
  10. A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
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    [Crossref] [PubMed]
  16. X. Y. Lü, P. J. Song, J. B. Liu, and X. X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2009).
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    [Crossref]
  22. J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
    [Crossref] [PubMed]
  23. I. L. Chuang and Y. Yamamot, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995).
    [Crossref] [PubMed]
  24. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
    [Crossref]
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    [Crossref] [PubMed]
  26. M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
    [Crossref]
  27. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
    [Crossref]
  28. Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
    [Crossref]
  29. S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
    [Crossref]
  30. L. L. Sun, H. F. Wang, S. Zhang, and K. H. Yeon, “Entanglement concentration of partially entangled three-photon W states with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 630–634 (2012).
    [Crossref]
  31. S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
    [Crossref]
  32. X. H. Li and S. Ghose, “Efficient hyperconcentration of nonlocal multipartite entanglement via the cross-Kerr nonlinearity,” Opt. Express 23, 3550–3562 (2015).
    [Crossref] [PubMed]
  33. N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985).
    [Crossref]
  34. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
    [Crossref]
  35. M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
    [Crossref] [PubMed]
  36. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
    [Crossref]
  37. T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
    [Crossref]
  38. S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
    [Crossref] [PubMed]
  39. H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
    [Crossref]
  40. H. Jeong, “Quantum computation using weak nonlinearities: Robustness against decoherence,” Phys. Rev. A 73, 052320 (2006).
    [Crossref]
  41. L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
    [Crossref]
  42. R. W. Boyd, “Order-of-magnitude estimates of the nonlinear optical susceptibility,” J. Mod. Opt. 46, 367–378 (1999).
    [Crossref]
  43. P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
    [Crossref]
  44. E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
    [Crossref] [PubMed]
  45. J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
    [Crossref]

2016 (1)

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

2015 (1)

2014 (2)

B. C. Ren and G. L. Long, “General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities,” Opt. Express 22, 6547–6561 (2014).
[Crossref] [PubMed]

N. K. Yu, C. Guo, and R. Y. Duan, “Obtaining a W state from a Greenberger-Horne-Zeilinger state via Stochastic local operations and classical communication with a rate approaching unity,” Phys. Rev. Lett. 112, 160401 (2014).
[Crossref] [PubMed]

2013 (3)

H. Q. Ma, K. J. Wei, and J. H. Yang, “Experimental circular quantum secret sharing over telecom fiber network,” Opt. Express 21, 16663–16669 (2013).
[Crossref] [PubMed]

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
[Crossref]

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
[Crossref]

2012 (4)

2011 (2)

2009 (2)

T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
[Crossref] [PubMed]

X. Y. Lü, P. J. Song, J. B. Liu, and X. X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2009).
[Crossref]

2007 (1)

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
[Crossref]

2006 (3)

T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
[Crossref]

H. Jeong, “Quantum computation using weak nonlinearities: Robustness against decoherence,” Phys. Rev. A 73, 052320 (2006).
[Crossref]

A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
[Crossref]

2005 (4)

F. L. Yan and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
[Crossref]

P. Walther, K. J. Resch, and A. Zeilinger, “Local conversion of Greenberger-Horne-Zeilinger states to approximate W states,” Phys. Rev. Lett. 94, 240501 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[Crossref]

H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
[Crossref]

2004 (2)

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[Crossref]

Q. Y. Cai and B. W. Li, “Improving the capacity of the Boström-Felbinger protocol,” Phys. Rev. A 69, 054301 (2004).
[Crossref]

2003 (1)

2002 (2)

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[Crossref] [PubMed]

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
[Crossref]

2001 (5)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[Crossref] [PubMed]

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

Y. S. Zhang, C. F. Li, and G. C. Guo, “Quantum key distribution via quantum encryption,” Phys. Rev. A 64, 024302 (2001).
[Crossref]

A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
[Crossref] [PubMed]

2000 (3)

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

J. Zhang and K. C. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref] [PubMed]

1999 (3)

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

R. W. Boyd, “Order-of-magnitude estimates of the nonlinear optical susceptibility,” J. Mod. Opt. 46, 367–378 (1999).
[Crossref]

1997 (2)

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

A. Zeilinger, M. A. Horne, H. Weinfurter, and M. Żukowski, “Three-Particle Entanglements from Two Entangled Pairs,” Phys. Rev. Lett. 78, 3031–3034 (1997).
[Crossref]

1995 (1)

I. L. Chuang and Y. Yamamot, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995).
[Crossref] [PubMed]

1992 (1)

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

1990 (1)

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
[Crossref] [PubMed]

1985 (1)

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985).
[Crossref]

Acín, A.

A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
[Crossref] [PubMed]

Armen, M. A.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[Crossref] [PubMed]

Au, J. K.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[Crossref] [PubMed]

Bai, J.

Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[Crossref]

Berthiaume, A.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

Bouwmeester, D.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

Boyd, R. W.

R. W. Boyd, “Order-of-magnitude estimates of the nonlinear optical susceptibility,” J. Mod. Opt. 46, 367–378 (1999).
[Crossref]

Bruß, D.

A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
[Crossref] [PubMed]

Bužek, V.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

Cai, Q. Y.

Q. Y. Cai and B. W. Li, “Improving the capacity of the Boström-Felbinger protocol,” Phys. Rev. A 69, 054301 (2004).
[Crossref]

Carri, J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref] [PubMed]

Cheng, L. Y.

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
[Crossref]

Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

Chuang, I. L.

I. L. Chuang and Y. Yamamot, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995).
[Crossref] [PubMed]

Cirac, J. I.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Daniell, M.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[Crossref] [PubMed]

Doherty, A. C.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[Crossref] [PubMed]

Dong, H. K.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

Dong, L.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

X. M. Xiu, L. Dong, Y. J. Gao, and X. X. Yi, “Quantum key distribution with Einstein-Podolsky-Rosen pairs associated with weak cross-Kerr nonlinearities,” J. Opt. Soc. Am. B 29, 2869–2874 (2012).
[Crossref]

Dowling, J. P.

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
[Crossref]

Duan, R. Y.

N. K. Yu, C. Guo, and R. Y. Duan, “Obtaining a W state from a Greenberger-Horne-Zeilinger state via Stochastic local operations and classical communication with a rate approaching unity,” Phys. Rev. Lett. 112, 160401 (2014).
[Crossref] [PubMed]

Dür, W.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Eibl, M.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

Fan, Q. B.

A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
[Crossref]

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

Gao, T.

F. L. Yan and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
[Crossref]

Gao, Y. J.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

X. M. Xiu, L. Dong, Y. J. Gao, and X. X. Yi, “Quantum key distribution with Einstein-Podolsky-Rosen pairs associated with weak cross-Kerr nonlinearities,” J. Opt. Soc. Am. B 29, 2869–2874 (2012).
[Crossref]

Gasparoni, S.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[Crossref] [PubMed]

Ghose, S.

Guo, C.

N. K. Yu, C. Guo, and R. Y. Duan, “Obtaining a W state from a Greenberger-Horne-Zeilinger state via Stochastic local operations and classical communication with a rate approaching unity,” Phys. Rev. Lett. 112, 160401 (2014).
[Crossref] [PubMed]

Guo, G. C.

Guo, Q.

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
[Crossref]

Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

Harris, S. E.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

Hau, L. V.

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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
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N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985).
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T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
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H. Jeong, “Quantum computation using weak nonlinearities: Robustness against decoherence,” Phys. Rev. A 73, 052320 (2006).
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H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
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E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
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P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
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T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
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P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
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A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
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Q. Y. Cai and B. W. Li, “Improving the capacity of the Boström-Felbinger protocol,” Phys. Rev. A 69, 054301 (2004).
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Y. S. Zhang, C. F. Li, and G. C. Guo, “Quantum key distribution via quantum encryption,” Phys. Rev. A 64, 024302 (2001).
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Li, Q. Y.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
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Liu, J. B.

Long, G. L.

Loock, P. V.

T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
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Lukin, M. D.

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
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M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
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Mabuchi, H.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
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D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

Mu, Q. X.

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
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T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
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T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
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W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
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K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
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L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
[Crossref] [PubMed]

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J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[Crossref] [PubMed]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

Peng, K. C.

J. Zhang and K. C. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
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S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
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T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
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Polzik, E. S.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
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J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
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Resch, K. J.

P. Walther, K. J. Resch, and A. Zeilinger, “Local conversion of Greenberger-Horne-Zeilinger states to approximate W states,” Phys. Rev. Lett. 94, 240501 (2005).
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A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of Mixed Three-Qubit States,” Phys. Rev. Lett. 87, 040401 (2001).
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Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
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Shapiro, J. H.

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
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Shen, H. Z.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
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J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
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Song, J.

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
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Song, P. J.

Spiller, T. P.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[Crossref]

Stockton, J. K.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[Crossref] [PubMed]

Su, S. L.

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
[Crossref]

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
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Sun, X. D.

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
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W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
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P. Walther, K. J. Resch, and A. Zeilinger, “Local conversion of Greenberger-Horne-Zeilinger states to approximate W states,” Phys. Rev. Lett. 94, 240501 (2005).
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Wang, H. F.

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
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L. L. Sun, H. F. Wang, S. Zhang, and K. H. Yeon, “Entanglement concentration of partially entangled three-photon W states with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 630–634 (2012).
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S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
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Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

H. F. Wang, S. Zhang, A. D. Zhu, X. X. Yi, and K. H. Yeon, “Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors,” Opt. Express 19, 25433–25440 (2011).
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Wang, J. X.

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
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Wei, K. J.

Weihs, G.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
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A. Zeilinger, M. A. Horne, H. Weinfurter, and M. Żukowski, “Three-Particle Entanglements from Two Entangled Pairs,” Phys. Rev. Lett. 78, 3031–3034 (1997).
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D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

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Xia, Y.

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
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A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
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L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
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T. Tashima, T. Wakatsuki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Local transformation of two Einstein-Podolsky-Rosen photon pairs into a three-photon W state,” Phys. Rev. Lett. 102, 130502 (2009).
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T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
[Crossref]

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985).
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Yan, F. L.

F. L. Yan and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
[Crossref]

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Yang, X. X.

Ye, L.

Yeon, K. H.

Yi, X. X.

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N. K. Yu, C. Guo, and R. Y. Duan, “Obtaining a W state from a Greenberger-Horne-Zeilinger state via Stochastic local operations and classical communication with a rate approaching unity,” Phys. Rev. Lett. 112, 160401 (2014).
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Zeilinger, A.

P. Walther, K. J. Resch, and A. Zeilinger, “Local conversion of Greenberger-Horne-Zeilinger states to approximate W states,” Phys. Rev. Lett. 94, 240501 (2005).
[Crossref]

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[Crossref] [PubMed]

A. Zeilinger, M. A. Horne, H. Weinfurter, and M. Żukowski, “Three-Particle Entanglements from Two Entangled Pairs,” Phys. Rev. Lett. 78, 3031–3034 (1997).
[Crossref]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Opt. Express 390, 575–579 (1997).

Zhang, J.

J. Zhang and K. C. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

Zhang, L. L.

J. Song, X. D. Sun, Q. X. Mu, L. L. Zhang, Y. Xia, and H. S. Song, “Direct conversion of a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process,” Phys. Rev. A 88, 024305 (2013).
[Crossref]

Zhang, S.

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
[Crossref]

L. L. Sun, H. F. Wang, S. Zhang, and K. H. Yeon, “Entanglement concentration of partially entangled three-photon W states with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 630–634 (2012).
[Crossref]

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
[Crossref]

Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

H. F. Wang, S. Zhang, A. D. Zhu, X. X. Yi, and K. H. Yeon, “Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors,” Opt. Express 19, 25433–25440 (2011).
[Crossref]

A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
[Crossref]

Zhang, Y. S.

Y. S. Zhang, C. F. Li, and G. C. Guo, “Quantum key distribution via quantum encryption,” Phys. Rev. A 64, 024302 (2001).
[Crossref]

Zheng, A. S.

Zhu, A. D.

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
[Crossref]

H. F. Wang, S. Zhang, A. D. Zhu, X. X. Yi, and K. H. Yeon, “Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors,” Opt. Express 19, 25433–25440 (2011).
[Crossref]

A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
[Crossref]

Zhu, L.

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
[Crossref]

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A. Zeilinger, M. A. Horne, H. Weinfurter, and M. Żukowski, “Three-Particle Entanglements from Two Entangled Pairs,” Phys. Rev. Lett. 78, 3031–3034 (1997).
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J. Opt. Soc. Am. B (3)

Nature (London) (1)

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature (London) 413, 273–276 (2001).
[Crossref]

New J. Phys. (2)

T. D. Ladd, P. V. Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006).
[Crossref]

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
[Crossref]

Opt. Commun. (2)

S. L. Su, L. Y. Cheng, H. F. Wang, and S. Zhang, “An economic and feasible scheme to generate the four-photon entangled state via weak cross-Kerr nonlinearity,” Opt. Commun. 293, 172–176 (2013).
[Crossref]

S. L. Su, L. Zhu, Q. Guo, H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Complete Bell-state and Greenberger-Horne-Zeilinger-state nondestructive detection based on simplified symmetry analyzer,” Opt. Commun. 285, 4134–4139 (2012).
[Crossref]

Opt. Express (7)

Phys. Rev. A (18)

Q. Guo, J. Bai, L. Y. Cheng, X. Q. Shao, H. F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[Crossref]

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
[Crossref] [PubMed]

H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
[Crossref]

H. Jeong, “Quantum computation using weak nonlinearities: Robustness against decoherence,” Phys. Rev. A 73, 052320 (2006).
[Crossref]

L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93, 012308 (2016).
[Crossref]

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66, 063814 (2002).
[Crossref]

J. Zhang and K. C. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

Y. S. Zhang, C. F. Li, and G. C. Guo, “Quantum key distribution via quantum encryption,” Phys. Rev. A 64, 024302 (2001).
[Crossref]

Q. Y. Cai and B. W. Li, “Improving the capacity of the Boström-Felbinger protocol,” Phys. Rev. A 69, 054301 (2004).
[Crossref]

A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, “Secure direct communication based on secret transmitting order of particles,” Phys. Rev. A 73, 022338 (2006).
[Crossref]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
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Figures (5)

Fig. 1
Fig. 1 Cross-Kerr nonlinear interaction model.
Fig. 2
Fig. 2 Schematic diagram for conversion from a GHZ state to a W state. And the qubit flip (QF) device is composed of two HV-PBS and a phase shifter (PS). Here, the FS-PBS, which transmits F-polarized photons and reflects S-polarized photons is the polarizing beam splitter (PBS) in the |F〉 and |S〉 basis. HV-PBS, which transmits H-polarized photons and reflects V-polarized photons, is the PBS in the in the |H〉 and |V〉 basis. PS denotes a phase shifter, which is to realize the transformation: |H〉 → |H〉, |V〉 → −|V〉.
Fig. 3
Fig. 3 (a) The effects of the homodyne measurement on the fidelity of the W state versus the amplitude of coherent state α and the distance between the measurement results and the phase shift angle θ, here we set δ = 10. (b) The effects of the homodyne measurement on the fidelity of the W state versus the amplitude of coherent state α and the distance between the measurement results and the peak δ, here we set θ = 0.01. (c) The effects of the homodyne measurement on the fidelity of the W state versus the phase shift angle θ and the distance between the measurement results and the peak δ, here we set α = 40000.
Fig. 4
Fig. 4 Curves of the homodyne measurement result X with α = 40000 and θ = 0.01.
Fig. 5
Fig. 5 (a) The measurement error rate versus the decoherence coefficient A and the amplitude α of the coherent state, here we set θ = 0.01. (b) The fidelity of the output state versus the decoherence coefficient A and the amplitude α of the coherent state, where we set the δ = 0.1d1, θ = 0.01.

Equations (13)

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H ^ QND = h ¯ χ a ^ s a ^ s a ^ p a ^ p .
U ck | Φ | α = e i H ^ QND t / h ¯ ( a | 0 + b | 1 ) | α = a | 0 | α + b | 1 | α e i θ .
| ϕ 0 = 1 2 ( | H 1 | H 2 | H 3 | H 4 + | V 1 | V 2 | V 3 | V 4 ) .
| ϕ 0 = 1 2 2 ( | F 1 | F 2 | F 3 | F 4 + | F 1 | S 2 | F 3 | S 4 + | S 1 | F 2 | F 3 | S 4 + | S 1 | S 2 | F 3 | F 4 + | F 1 | F 2 | S 3 | S 4 + | F 1 | S 2 | S 3 | F 4 + | S 1 | F 2 | S 3 | F 4 + | S 1 | S 2 | S 3 | S 4 ) | α .
| ϕ 2 = 1 2 2 [ ( | S a | F b | F c | F d + | F a | S b | F c | S d + | S a | F b | S c | S d + | F a | F b | F c | S d ) + | α e 3 θ + ( | F a | S b | S c | S d + | S a | F b | F c | F d + | S a | S b | F c | S d + | S a | S b | S c | F d ) | α e θ ] .
| ϕ X X | ϕ 3 = 1 N [ 2 f ( x , α cos 3 θ ) e i ϕ ( X ) 1 | φ 1 + 2 f ( x , α cos θ ) e i φ ( X ) 2 | φ 2 ] ,
f ( x , β ) = exp [ 1 4 ( x 2 β ) 2 ] / ( 2 π ) 1 / 4 , ϕ ( X ) 1 = α sin 3 θ ( x 2 α cos 3 θ ) Mod 2 π , ϕ ( X ) 2 = α sin θ ( x 2 α cos θ ) Mod 2 π , | φ 1 = 1 2 ( | S a | F b | F c | F d + | F a | S b | F c | S d + | F a | F b | S c | F d + | F a | F b | F c | S d ) , | φ 2 = 1 2 ( | F a | S b | S c | S d + | S a | F b | S c | S d + | S a | S b | F c | S d + | S a | S b | S c | F d ) , N [ 4 f 2 ( x , α cos 3 θ ) + 4 f 2 ( x , α cos θ ) ] 1 / 2 .
F ( X 1 ) = | φ | ϕ 1 X | 1 1 + e 2 d 1 ( d 1 δ 1 ) ,
ρ t = J ^ ρ + L ^ ρ , J ^ ρ = γ a ρ a , L ^ ρ = γ 2 ( a a ρ + ρ a a ) .
ρ ( t ) = [ D ˜ ( Δ t ) U ˜ ( Δ t ) ] N ρ ( 0 ) = j , k = 1 , 3 e B j , k ( | ϕ j ϕ k | ) | A a e i j θ A a e i k θ | ,
A = γ t / 2 , B j , k = α 2 ( 1 A 2 N ) n = 1 N A 2 ( n 1 ) N [ e i ( J k ) n θ N 1 ] .
| x ρ ( t ) x | = j , k = 1 , 3 C j , k ( | ϕ j ϕ k | ) f ( x , j θ ) f ( x , k θ ) e i [ μ j k + φ ( x , j θ ) φ ( x , k θ ) ] ,
C j , k = e Re ( B j k ) , μ j k = Im ( B j k ) , f ( x , n θ ) = exp [ 1 4 ( x 2 A α cos n θ ) 2 ) ] / ( 2 π ) 1 / 4 ( n = 1 , 3 ) , φ ( x , n θ ) = ( x 2 A α cos n θ ) A α sin n θ ( n = 1 , 3 ) .

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