Abstract

Hyperentanglement involves multiple degrees of freedom of a quantum system and has attracted a lot of attention recently because of its high efficiency in quantum applications. We propose some practical schemes using linear optics for partially entangled n-photon 2n-qubit systems with spatial and polarization degrees of freedom. The states involved are not equivalent to the general Bell states or GHz states under local quantum operations and classical communication. Our schemes are based on the parameter-splitting method, which can change different entanglement coefficients into equal coefficients. They are very efficient and practical as they use only linear-optical elements and do not require nonlinear optics.

© 2013 Optical Society of America

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  38. Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
    [CrossRef]
  39. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
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  40. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).
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  43. B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser Phys. Lett. 10, 115201 (2013).
    [CrossRef]
  44. T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069–1076 (2013).
    [CrossRef]
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    [CrossRef]
  46. B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
    [CrossRef]
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    [CrossRef]
  48. S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
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  49. C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
    [CrossRef]
  50. M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
    [CrossRef]
  51. S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
    [CrossRef]
  52. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
    [CrossRef]
  53. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
    [CrossRef]
  54. Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
    [CrossRef]
  55. B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
    [CrossRef]
  56. T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
    [CrossRef]
  57. T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
    [CrossRef]
  58. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [CrossRef]
  59. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
    [CrossRef]

2013 (5)

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
[CrossRef]

B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[CrossRef]

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser Phys. Lett. 10, 115201 (2013).
[CrossRef]

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069–1076 (2013).
[CrossRef]

T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
[CrossRef]

2012 (7)

B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
[CrossRef]

T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
[CrossRef]

F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
[CrossRef]

B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012).
[CrossRef]

F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multiphoton system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012).
[CrossRef]

2011 (4)

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[CrossRef]

F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
[CrossRef]

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
[CrossRef]

W. Xiong and L. Ye, “Schemes for entanglement concentration of two unknown partially entangled states with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 28, 2030–2037 (2011).
[CrossRef]

2010 (7)

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B 27, 2159–2164 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[CrossRef]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[CrossRef]

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D 56, 271–275 (2010).
[CrossRef]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

2009 (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[CrossRef]

2008 (3)

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric down conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[CrossRef]

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

2007 (1)

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[CrossRef]

2006 (1)

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[CrossRef]

2005 (2)

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

2004 (1)

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[CrossRef]

2003 (5)

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[CrossRef]

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
[CrossRef]

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[CrossRef]

S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
[CrossRef]

2002 (4)

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
[CrossRef]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
[CrossRef]

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

X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[CrossRef]

2001 (3)

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[CrossRef]

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
[CrossRef]

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
[CrossRef]

2000 (2)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef]

B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
[CrossRef]

1999 (3)

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[CrossRef]

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

A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999).
[CrossRef]

1998 (1)

P. G. Kwiat and H. Weinfurter, “Embedded Bell-state analysis,” Phys. Rev. A 58, R2623–R2626 (1998).
[CrossRef]

1997 (1)

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

1996 (3)

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

1993 (1)

C. H. Bennett, G. Brassard, C. Cr’epeau, 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]

1992 (1)

C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef]

Barbieri, M.

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[CrossRef]

Barreiro, J. T.

T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
[CrossRef]

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, C. Cr’epeau, 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]

C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef]

Bernstein, H. J.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Berthiaume, A.

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

Bose, S.

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[CrossRef]

Bouwmeester, D.

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

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, C. Cr’epeau, 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).
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J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
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M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
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Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
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M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

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C. H. Bennett, G. Brassard, C. Cr’epeau, 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).
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M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[CrossRef]

Deng, F. G.

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser Phys. Lett. 10, 115201 (2013).
[CrossRef]

B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[CrossRef]

F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multiphoton system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012).
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F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
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B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
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F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
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F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
[CrossRef]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[CrossRef]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric down conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[CrossRef]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
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D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

Du, F. F.

Eibl, M.

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

Ekert, A.

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

Feng, L.

X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[CrossRef]

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J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Gisin, N.

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

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T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
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Gu, B.

Guo, G. C.

B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
[CrossRef]

Hillery, M.

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

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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Hua, M.

Huber, G.

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
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T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
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A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999).
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T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef]

Jiang, Y. K.

B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
[CrossRef]

Jin, G. S.

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[CrossRef]

Jozsa, R.

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, C. Cr’epeau, 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]

Karlsson, A.

A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999).
[CrossRef]

Knight, P. L.

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[CrossRef]

Koashi, M.

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
[CrossRef]

A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999).
[CrossRef]

Kurtsiefer, C.

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[CrossRef]

Kwiat, P. G.

T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
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J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
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J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
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P. G. Kwiat and H. Weinfurter, “Embedded Bell-state analysis,” Phys. Rev. A 58, R2623–R2626 (1998).
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J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Li, T.

Li, X. H.

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[CrossRef]

Li, Y. S.

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

Liu, X. S.

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[CrossRef]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
[CrossRef]

X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[CrossRef]

Long, G. L.

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069–1076 (2013).
[CrossRef]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
[CrossRef]

T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[CrossRef]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[CrossRef]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[CrossRef]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
[CrossRef]

X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[CrossRef]

Lu, Y.

T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
[CrossRef]

Macchiavello, C.

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef]

Mataloni, P.

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[CrossRef]

Mattle, K.

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

Mohseni, M.

T. M. Graham, J. T. Barreiro, M. Mohseni, and P. G. Kwiat, “Hyperentanglement-enabled direct characterization of quantum dynamics,” Phys. Rev. Lett. 110, 060404 (2013).
[CrossRef]

Monken, C. H.

S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
[CrossRef]

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Nogueira, W. A. T.

S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
[CrossRef]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef]

Padua, S.

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[CrossRef]

S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
[CrossRef]

Pan, J. W.

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[CrossRef]

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
[CrossRef]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
[CrossRef]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[CrossRef]

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
[CrossRef]

Pan, J.-W.

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

Peres, A.

C. H. Bennett, G. Brassard, C. Cr’epeau, 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]

Peters, N. A.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Popescu, S.

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef]

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M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Ren, B. C.

B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[CrossRef]

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser Phys. Lett. 10, 115201 (2013).
[CrossRef]

F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multiphoton system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012).
[CrossRef]

B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012).
[CrossRef]

Ribordy, G.

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

Sanpera, A.

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef]

Schuck, C.

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[CrossRef]

Schumacher, B.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

Sheng, Y. B.

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
[CrossRef]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric down conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

Shi, B. S.

B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
[CrossRef]

Simon, C.

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
[CrossRef]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[CrossRef]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef]

Smolin, J. A.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

Tittel, W.

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

Tong, D. M.

X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65, 022304 (2002).
[CrossRef]

Ursin, R.

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Vallone, G.

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[CrossRef]

Vedral, V.

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
[CrossRef]

Walborn, S. P.

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[CrossRef]

S. P. Walborn, W. A. T. Nogueira, S. Padua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. 62, 161 (2003).
[CrossRef]

Wang, C.

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[CrossRef]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[CrossRef]

Wang, H. F.

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B 27, 2159–2164 (2010).
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T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069–1076 (2013).
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T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
[CrossRef]

Wei, H. R.

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

Weihs, G.

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C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
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T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
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C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef]

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C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

C. H. Bennett, G. Brassard, C. Cr’epeau, 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).
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L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
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Xiong, W.

Yamamoto, T.

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
[CrossRef]

Yang, T.

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
[CrossRef]

Ye, L.

Yeon, K. H.

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B 27, 2159–2164 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D 56, 271–275 (2010).
[CrossRef]

Zbinden, H.

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

Zeilinger, A.

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[CrossRef]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef]

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

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[CrossRef]

Zellinger, A.

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Zhan, M. S.

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
[CrossRef]

Zhang, A. N.

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
[CrossRef]

Zhang, S.

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B 27, 2159–2164 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D 56, 271–275 (2010).
[CrossRef]

Zhang, Y.

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[CrossRef]

Zhao, S. M.

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
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Zhao, Z.

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
[CrossRef]

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
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Zheng, B. Y.

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Zhou, H. Y.

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric down conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
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Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[CrossRef]

Zhou, L.

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
[CrossRef]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Eur. Phys. J. D (1)

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D 56, 271–275 (2010).
[CrossRef]

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Nat. Phys. (1)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008).
[CrossRef]

Nature (3)

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

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[CrossRef]

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (30)

T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012).
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[CrossRef]

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[CrossRef]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric down conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[CrossRef]

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[CrossRef]

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011).
[CrossRef]

F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
[CrossRef]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
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[CrossRef]

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
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[CrossRef]

F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
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S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
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L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
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G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
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F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
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C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
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Phys. Rev. Lett. (12)

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef]

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[CrossRef]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
[CrossRef]

C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef]

C. H. Bennett, G. Brassard, C. Cr’epeau, 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]

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[CrossRef]

Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
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J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
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Quantum Inf. Comput. (1)

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

Rev. Mod. Phys. (2)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic photon circuit of our HECP for known partially hyperentangled W state. (b) Schematic photon circuit of an unbalanced BS (UBS), see [30]. The BS represents a balanced 5050 beam splitter. The phS represents a phase shift that decides the reflection coefficient and transmission coefficient of the UBS. x1 and x2 represent two original spatial modes of photon X (X=A, B). x1 and x2 are another two spatial modes of photon X. UBSx represents an unbalanced beam splitter with reflection coefficient R1=α2/γ2 or R2=α2/β2 for X=A, B, respectively. Rx(θ) represents a wave plate rotating the horizontal polarization with angle θ=arccos(α1/β1) or θ=arccos(α1/γ1) for Bob and Alice, respectively. PBSxi (i=1,2,,6) represent a polarizing beam splitter, which transmits |H and reflects |V. DL denotes a time-delay device used to make two wave packets reach the last PBS (PBSx5 or PBSx6) in each Mach–Zehnder interferometer simultaneously. D1, D2, and D3 represent three single-photon detectors.

Fig. 2.
Fig. 2.

Success probability P1 of our HECP for a partially hyperentangled three-photon W state with known parameters. It only depends on the minimal entanglement coefficients α1 and α2.

Fig. 3.
Fig. 3.

Success probability P2 of the HECP for partially hyperentangled three-photon W-type state with known parameters. P2 only depends on the minimal entanglement coefficients α0, β0.

Fig. 4.
Fig. 4.

Schematic diagram of the HECP for partially entangled W-cluster states with known parameters. UBSxi represent unbalanced beam splitters with reflection coefficients RXi, X=A, B, C, i=1, 2, 3, 4. DL denotes a time-delay device that is used to make two wave packets reach the last PBS in each Mach–Zehnder interferometer simultaneously.

Equations (26)

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

|ϕ0ABC=(α1|HHV+β1|HVH+γ1|VHH)ABC(α2|a1b1c2+β2|a1b2c1+γ2|a2b1c1).
|ϕ1ABC=(α1|HHV+β1|HVH+γ1|VHH)ABC(α2|a1b1c2+α2|a1b2c1+γ2|a2b1c1+β2|a1b3c1)
|ϕ2ABC=(α1|HHV+β1|HVH+γ1|VHH)ABC(α2(|a1b1c2+|a1b2c1+|a2b1c1)+β2|a1b3c1+γ2|a3b3c1)
|ϕ3ABC=(α1(|HHV+|HVH+|VHH)+β1|HVH+γ1|VHH)α2(|a1b1c2+|a1b2c1+|a2b1c1)+(α1|HHV+β1|HVH+γ1|VHH)(β2|a1b3c1+γ2|a3b1c1)
|ϕ4ABC=α1α2(|HHV+|HVH+|VHH)(|a1b1c2+|a1b2c1+|a2b1c1)+α2(β1|HVH+γ1|VHH)(|a1b1c2+|a1b2c1+|a2b1c1)+(α1|HHV+β1|HVH+γ1|VHH)(β2|a1b3c1+γ2|a3b1c1).
|ϕABC=12(|HHV+|HVH+|VHH)(|a1b1c2+|a1b2c1+|a2b1c1).
|φ0ABC=(α0|HHH+α1|HHV+α2|HVH+α3|VHH)(β0|a1b1c1+β1|a1b1c2+β2|a1b2c1+β3|a2b1c1).
(α0|HHH+α1|HHV+α2|HVH+α3|VHH)(β0|a1b1c1+β0|a1b1c2+β0|a1b2c1+β0|a2b1c1)+(α0|HHH+α1|HHV+α2|HVH+α3|VHH)(β1|a1b1c3+β2|a1b3c1+β3|a3b1c1)
α0β0(|HHH+|HHV+|HVH+|VHH)(|a1b1c1+|a1b1c2+|a1b2c1+|a2b1c1)+(α1|HHV+α2|HVH+α3|VHH)β0(|a1b1c1+|a1b1c2+|a1b2c1+|a2b1c1)+(α0|HHH+α1|HHV+α2|HVH+α3|VHH)(β1|a1b1c3+β2|a1b3c1+β3|a3b1c1)
α0β0(|HHH+|HHV+|HVH+|VHH)(|a1b1c1+|a1b1c2+|a1b2c1+|a2b1c1)+(α1|HHV+α2|HVH+α3|VHH)β0(|a1b1c1+|a1b1c2+|a1b2c1+|a2b1c1)+(α0|HHH+α1|HHV+α2|HVH+α3|VHH)(β1|a1b1c3+β2|a1b3c1+β3|a3b1c1).
|φABC=14(|HHH+|HHV+|HVH+|VHH)(|a1b1c1+|a1b1c2+|a1b2c1+|a2b1c1).
|Φ0A1An=(i=1nαi|Wi)(i=1nβi|wi).
i=1n|αi|2=i=1n|βi|2=1.
|Φ1A1An=(i=1nαi|Wi)(i=1nβ1|wi)(i=1nαi|Wi)(i=2nβ^i|wi)
|Φ2A1An=α1β1(i=1n|Wi)(i=1n|wi)+(i=2nα^i|Wi)(i=1n|wi)+(i=1nαi|Wi)(i=2nβ^i|wi)
|Φ3A1An=α1β1(i=1n|Wi)(i=1n|wi)+(i=2nα^i|Wi)(i=1n|wi)+(i=1nαi|Wi)(i=2nβ^i|wi).
|ΦA1An=1n(i=1n|Wi)(i=1n|wi).
|Ψ0A1An=(i=0nαi|Wi)(i=0nβi|wi).
|ΨA1An=1n+1(i=0n|Wi)(i=0n|wi),
|ψ0ABC=α1|HHV|a1b1c2+α2|HVH|a1b1c2+α3|VHH|a1b1c2+α4|HHV|a1b2c1+α5|HVH|a1b2c2+α6|VHH|a1b2c2+α7|HHV|a2b1c1+α8|HVH|a2b1c1α9|VHH|a2b1c1.
|ψ1ABC=α1x1|HHV|a1b1c2+α2x1|HVH|a1b1c2+α3x2|VHH|a1b1c2+α4x1|HHV|a1b2c1+α5x1|HVH|a1b2c2+α6x2|VHH|a1b2c2+α7x3|HHV|a2b1c1+α8x3|HVH|a2b1c1α9x4|VHH|a2b1c1+α1x1|HHV|a3b1c2+α2x1|HVH|a3b1c2+α4x1|HHV|a3b2c1+α5x1|HVH|a3b2c2+α3x2|VHH|a4b1c2+α6x2|VHH|a4b2c2+α7x3|HHV|a5b1c1+α8x3|HVH|a2b1c1α9x4|VHH|a2b1c1
|ψ2ABC=α1x1y1|HHV|a1b1c2+α2x1y2|HVH|a1b1c2+α3x2y1|VHH|a1b1c2+α4x1y3|HHV|a1b2c1+α5x1y4|HVH|a1b2c2+α6x2y3|VHH|a1b2c2+α7x3y1|HHV|a2b1c1+α8x3y2|HVH|a2b1c1α9x4y1|VHH|a2b1c1+α1x1y1|HHV|a3b1c2+α2x1y2|HVH|a3b1c2+α4x1y3|HHV|a3b2c1+α5x1y4|HVH|a3b2c2+α3x2y1|VHH|a4b1c2+α6x2y3|VHH|a4b2c2+α7x3y1|HHV|a5b1c1+α8x3y2|HVH|a2b1c1α9x4y1|VHH|a2b1c1+α1y1x1|HHV|a1b3c2+α3y1x2|VHH|a1b3c2+α7y1x3|HHV|a2b3c1α9y1x4|VHH|a2b3c1+α1y1x1|HHV|a3b3c2+α3y1x2|VHH|a4b3c2+α7y1x3|HHV|a5b3c1α9y1x4|VHH|a2b3c1+α2x1y2|HVH|a1b4c2+α8x3y2|HVH|a2b4c1+α2y2x1|HVH|a3b4c2+α8y2x3|HVH|a2b4c1+α4x1y3|HHV|a1b5c1+α6x2y3|VHH|a1b5c2+α4y3x1|HHV|a3b5c1+α5y4x1|HVH|a3b6c2.
|ψ3ABC=α1x1y1z4|HHV|a1b1c2+α2x1y2z3|HVH|a1b1c2+α3x2y1z3|VHH|a1b1c2+α4x1y3z2|HHV|a1b2c1+α5x1y4z3|HVH|a1b2c2+α6x2y3z3|VHH|a1b2c2+α7x3y1z2|HHV|a2b1c1+α8x3y2z1|HVH|a2b1c1α9x4y1z1|VHH|a2b1c1+x1α1y1z4|HHV|a3b1c2+α2x1y2z3|HVH|a3b1c2+α4x1y3z2|HHV|a3b2c1+α5x1y4z3|HVH|a3b2c2+α3x2y1z3|VHH|a4b1c2+α6x2y3z3|VHH|a4b2c2+α7x3y1z2|HHV|a5b1c1+α8x3y2z1|HVH|a2b1c1α9x4y1z1|VHH|a2b1c1+α1y1x1z4|HHV|a1b3c2+α3y1x2z3|VHH|a1b3c2+α7y1x3z2|HHV|a2b3c1α9y1x4z1|VHH|a2b3c1+α1x1y1z4|HHV|a3b3c2+α3x2y1z3|VHH|a4b3c2+α7x3y1z2|HHV|a5b3c1α9x4y1z1|VHH|a2b3c1+α2x1y2z3|HVH|a1b4c2+α8x3y2z1|HVH|a2b4c1+α2x1y2z3|HVH|a3b4c2+α8x3y2z1|HVH|a2b4c1+α4x1y3z2|HHV|a1b5c1+α6x2y3z3|VHH|a1b5c2+α4x1y3z2|HHV|a3b5c1+α5x1y4z3|HVH|a3b6c2+α8x3y2z1|HVH|a2b1c1α9x4y1z1|VHH|a2b1c1+α8x3y2z1|HVH|a2b1c1α9x4y1z1|VHH|a2b1c1α9x4y1z1|VHH|a2b3c1+α9x4y1z1|VHH|a2b3c1+α8x3y2z1|HVH|a2b4c1+α8y2x3z1|HVH|a2b4c1+α4x1y3z2|HHV|a1b2c1+α7x3y1z2|HHV|a2b1c1+α4x1y3z2|HHV|a3b2c1+α7x3y1z2|HHV|a5b1c1+α7y1x3z2|HHV|a2b3c1+α7y1x3z2|HHV|a5b3c1+α4y3x1z2|HHV|a1b5c1+α4y3x1z2|HHV|a3b5c1+α2x1y2z3|HVH|a1b1c2+α5x1y4z3|HVH|a1b2c2+α6x2y3z3|VHH|a1b2c2+α2x1y2z3|HVH|a3b1c2+α5x1y4z3|HVH|a3b2c2+α3x2y1z3|VHH|a4b1c2+α6x2y3z3|VHH|a4b2c2+α3y1x2z3|VHH|a1b3c2+α3y1x2z3|VHH|a4b3c2+α2y2x1z3|HVH|a1b4c2+α2y2x1z3|HVH|a3b4c2+α6y3x2z3|VHH|a1b5c2+α5y4x1z3|HVH|a3b6c2+α1x1y1z4|HHV|a1b1c2+α1x1y1z4|HHV|a3b1c2+α1x1y1z4|HHV|a1b3c2+α1y1x1z4|HHV|a3b3c2.
α1x1y1z4=α2x1y2z3=α3x2y1z3=α4x1y3z2=α5x1y4z3=α6x2y3z3=α7x3y1z2=α8x3y2z1=α9x4y1z1.
|φABC=13(|HHV|a1b1c2+|HVH|a1b1c2+|VHH|a1b1c2+|HHV|a1b2c1+|HVH|a1b2c2+|VHH|a1b2c2+|HHV|a2b1c1+|HVH|a2b1c1|VHH|a2b1c1).
x2=δα2α6α7α32α4α8y1z1,x3=α3α4x1y1α6α7y1x4=δα9y1z1,y2=δα6α7α3α4α8x1z1y3=α3y1α6,y4=kα2α6α7α3α4α5α8z1z2=δα6α3α4x1y1,z3=α3α3α8z1α2α6α7z4=δα1x1y1,δ=α1x1y1z4

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