Abstract

We propose a single-photon-assisted entanglement concentration protocol (ECP) for nonlocal N-photon systems in a partially entangled pure W-class state with linear optical elements. Only one of the N parties in quantum communication prepares an ancillary photon and operates the entanglement concentration process for picking up the standard N-photon W state from each partially entangled pure W-class state by choosing the two-mode instances from a polarization beam splitter. Compared with other ECPs for W-class states, our protocol has some advantages. First, it requires only linear optical elements. Second, it requires an N-photon system and an ancillary photon for each round of concentration, not two systems. Third, only one party asks other parties to retain or discard their photons. All these advantages make our ECP more feasible and more convenient than others.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  2. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [CrossRef]
  3. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
    [CrossRef]
  4. F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003).
    [CrossRef]
  5. X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
    [CrossRef]
  6. C. H. Bennett, G. Brassard, C. Crepeau, 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]
  7. F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
    [CrossRef]
  8. 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]
  9. 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]
  10. G. L. Long, and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002).
    [CrossRef]
  11. 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]
  12. F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (2004).
    [CrossRef]
  13. 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]
  14. F. L. Yan and X. Zhang, “A scheme for secure direct communication using EPR pairs and teleportation,” Eur. Phys. J. B 41, 75–78 (2004).
    [CrossRef]
  15. T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
    [CrossRef]
  16. T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
    [CrossRef]
  17. T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
    [CrossRef]
  18. Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
    [CrossRef]
  19. 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]
  20. X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
    [CrossRef]
  21. J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
    [CrossRef]
  22. C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (2005).
    [CrossRef]
  23. B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
    [CrossRef]
  24. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
    [CrossRef]
  25. A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999).
    [CrossRef]
  26. 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]
  27. F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (2005).
    [CrossRef]
  28. F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
    [CrossRef]
  29. Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
    [CrossRef]
  30. F. L. Yan, and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
    [CrossRef]
  31. F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
    [CrossRef]
  32. B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
    [CrossRef]
  33. B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
    [CrossRef]
  34. A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
    [CrossRef]
  35. F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
    [CrossRef]
  36. F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
    [CrossRef]
  37. X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
    [CrossRef]
  38. 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]
  39. 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]
  40. J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
    [CrossRef]
  41. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
    [CrossRef]
  42. 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]
  43. M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
    [CrossRef]
  44. 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]
  45. Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
    [CrossRef]
  46. F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
    [CrossRef]
  47. X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
    [CrossRef]
  48. F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
    [CrossRef]
  49. C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).
  50. C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685–25695 (2011).
    [CrossRef]
  51. Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
    [CrossRef]
  52. Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (2012).
    [CrossRef]
  53. B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
    [CrossRef]
  54. X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
    [CrossRef]
  55. Y. B. Sheng, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (2011).
    [CrossRef]
  56. M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
    [CrossRef]
  57. 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]
  58. S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999).
    [CrossRef]
  59. B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
    [CrossRef]
  60. T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
    [CrossRef]
  61. Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001).
    [CrossRef]
  62. M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
    [CrossRef]
  63. Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
    [CrossRef]
  64. 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]
  65. 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).
  66. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
    [CrossRef]
  67. 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]
  68. F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
    [CrossRef]
  69. 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]
  70. 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]
  71. 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]
  72. 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]
  73. 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]
  74. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
    [CrossRef]

2012

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (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]

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

2011

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]

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]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (2011).
[CrossRef]

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

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

C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685–25695 (2011).
[CrossRef]

2010

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (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 and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (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]

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

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

2009

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[CrossRef]

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[CrossRef]

2008

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]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (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]

2006

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]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[CrossRef]

J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[CrossRef]

2005

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
[CrossRef]

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
[CrossRef]

C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (2005).
[CrossRef]

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
[CrossRef]

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

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (2005).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (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]

T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
[CrossRef]

Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
[CrossRef]

2004

F. L. Yan and X. Zhang, “A scheme for secure direct communication using EPR pairs and teleportation,” Eur. Phys. J. B 41, 75–78 (2004).
[CrossRef]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef]

F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (2004).
[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]

2003

F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003).
[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]

2002

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]

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

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

2001

J. W. Pan, C. Simon, and A. Zellinger, “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

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

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

1999

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. Bužek, 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]

1996

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]

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]

1993

C. H. Bennett, G. Brassard, C. Crepeau, 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

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[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]

1991

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[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. Crepeau, 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, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[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]

Berthiaume, A.

M. Hillery, V. Bužek, 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]

Bowen, W. P.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[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. Crepeau, 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, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[CrossRef]

Bužek, V.

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

Cao, Z. L.

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[CrossRef]

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
[CrossRef]

Chen, Y. L.

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
[CrossRef]

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

Chuang, I. L.

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

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]

Crepeau, C.

C. H. Bennett, G. Brassard, C. Crepeau, 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]

Czechlewski, M.

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[CrossRef]

Deng, F. G.

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

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (2012).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (2011).
[CrossRef]

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (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 and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (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, “Efficient polarization entanglement concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[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, 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]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[CrossRef]

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (2005).
[CrossRef]

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (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]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (2004).
[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 and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003).
[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]

Deutsch, D.

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]

Ding, L. G.

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

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]

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]

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef]

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]

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]

Feng, X. L.

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
[CrossRef]

Gao, T.

T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
[CrossRef]

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

Grudka, A.

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[CrossRef]

Gu, B.

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
[CrossRef]

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[CrossRef]

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

Huang, Y. G.

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
[CrossRef]

Imoto, N.

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]

Ishizaka, S.

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[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, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

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. Crepeau, 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]

Kwek, L. C.

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
[CrossRef]

Lam, P. K.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef]

Lance, A. M.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef]

Li, C. Q.

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

Li, C. Y.

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

Li, X. H.

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

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[CrossRef]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

Li, Y.

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
[CrossRef]

Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
[CrossRef]

Li, Y. S.

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (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]

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.

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (2012).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (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]

C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (2005).
[CrossRef]

F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (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 and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (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]

F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003).
[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]

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

Man, Z. X.

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
[CrossRef]

Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
[CrossRef]

Mermin, N. D.

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[CrossRef]

Mu, L. L.

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

Nielsen, M. A.

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

Oh, C. H.

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
[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]

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

J. W. Pan, C. Simon, and A. Zellinger, “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]

Pei, S. X.

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[CrossRef]

Peres, A.

C. H. Bennett, G. Brassard, C. Crepeau, 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]

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]

Sanders, B. C.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[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]

Schumacher, B.

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]

Sheng, Y. B.

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (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, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (2011).
[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 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 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, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
[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, and A. Zellinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
[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]

Song, B.

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[CrossRef]

Song, W.

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
[CrossRef]

Sun, L. L.

Symul, T.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef]

Tang, C. J.

J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
[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]

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]

Vidal, G.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[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, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685–25695 (2011).
[CrossRef]

C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (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]

Wang, H. F.

Wang, J.

J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
[CrossRef]

Wang, T. J.

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[CrossRef]

Wang, Y.

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

Wang, Z. X.

T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
[CrossRef]

Wiesner, S. J.

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]

Wójcik, A.

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[CrossRef]

Wootters, W. K.

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. Crepeau, 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]

Xia, Y.

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]

Xiao, L.

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]

Xiong, W.

Xu, F.

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

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]

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]

T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
[CrossRef]

T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
[CrossRef]

F. L. Yan and X. Zhang, “A scheme for secure direct communication using EPR pairs and teleportation,” Eur. Phys. J. B 41, 75–78 (2004).
[CrossRef]

Yang, M.

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[CrossRef]

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
[CrossRef]

Ye, L.

Yeon, K. H.

Zellinger, A.

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

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
[CrossRef]

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

Zhang, L. H.

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[CrossRef]

Zhang, Q.

J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
[CrossRef]

Zhang, R.

Zhang, S.

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]

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]

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, X.

F. L. Yan and X. Zhang, “A scheme for secure direct communication using EPR pairs and teleportation,” Eur. Phys. J. B 41, 75–78 (2004).
[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]

C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685–25695 (2011).
[CrossRef]

Zhang, Z. J.

Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
[CrossRef]

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
[CrossRef]

Zhao, B. K.

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[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]

Zhao, Y.

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

Zhao, Z.

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

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]

Zhong, K.

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[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 concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[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, 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]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[CrossRef]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
[CrossRef]

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[CrossRef]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (2005).
[CrossRef]

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

Zhou, L.

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, P.

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[CrossRef]

Zhu, A. D.

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]

Chin. Phys.

T. Gao, F. L. Yan, and Z. X. Wang, “Controlled quantum teleportation and secure direct communication,” Chin. Phys. 14, 893–897 (2005).
[CrossRef]

Chin. Phys. B

B. Gu, C. Q. Li, F. Xu, and Y. L. Chen, “High-capacity three-party quantum secret sharing with superdense coding,” Chin. Phys. B 18, 4690–4694 (2009).
[CrossRef]

Chin. Phys. Lett.

Z. X. Man, Z. J. Zhang, and Y. Li, “Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations,” Chin. Phys. Lett. 22, 18–21 (2005).
[CrossRef]

B. Gu, Y. L. Chen, C. Y. Zhang, and Y. G. Huang, “Efficient polarization entanglement purification using spatial entanglement,” Chin. Phys. Lett. 27, 100304 (2010).
[CrossRef]

Eur. Phys. J. B

F. L. Yan and X. Zhang, “A scheme for secure direct communication using EPR pairs and teleportation,” Eur. Phys. J. B 41, 75–78 (2004).
[CrossRef]

Eur. Phys. J. D

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Quantum state sharing of an arbitrary two-qubit state with two-photon entanglements and Bell-state measurements,” Eur. Phys. J. D 39, 459–464 (2006).
[CrossRef]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[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]

Int. J. Mod. Phys. C

T. Gao, F. L. Yan, and Z. X. Wang, “Quantum secure conditional direct communication via EPR pairs,” Int. J. Mod. Phys. C 16, 1293 (2005).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A

T. Gao, F. L. Yan, and Z. X. Wang, “Deterministic secure direct communication using GHZ states and swapping quantum entanglement,” J. Phys. A 38, 5761–5770 (2005).
[CrossRef]

J. Phys. B

X. H. Li, P. Zhou, C. Y. Li, H. Y. Zhou, and F. G. Deng, “Efficient symmetric multiparty quantum state sharing of an arbitrary m-qubit state,” J. Phys. B 39, 1975–1983 (2006).
[CrossRef]

Nature

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

Opt. Commun.

B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, “Fault tolerant three-party quantum secret sharing against collective noise,” Opt. Commun. 283, 3099–3103 (2010).
[CrossRef]

C. Wang, F. G. Deng, and G. L. Long, “Multi-step quantum secure direct communication using multi-particle Green–Horne–Zeilinger state,” Opt. Commun. 253, 15–20 (2005).
[CrossRef]

Opt. Express

Phys. Lett. A

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314–319 (2012).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement concentration for electrons with charge detection,” Phys. Lett. A 373, 1823–1825 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, and G. L. Long, “Multipartite electronic entanglement purification with charge detection,” Phys. Lett. A 375, 396–400 (2011).
[CrossRef]

F. G. Deng, G. L. Long, and H. Y. Zhou, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337, 329–334 (2005).
[CrossRef]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Efficient high-capacity quantum secret sharing with two-photon entanglement,” Phys. Lett. A 372, 1957–1962 (2008).
[CrossRef]

J. Wang, Q. Zhang, and C. J. Tang, “Quantum secure direct communication based on order rearrangement of single photons,” Phys. Lett. A 358, 256–258 (2006).
[CrossRef]

Phys. Rev. A

M. Hillery, V. Bužek, 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]

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]

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 044301 (2005).
[CrossRef]

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

F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, “Improving the security of multiparty quantum secret sharing against Trojan horse attack,” Phys. Rev. A 72, 044302 (2005).
[CrossRef]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005).
[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]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006).
[CrossRef]

F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[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]

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

F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (2004).
[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]

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005).
[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]

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

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

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005).
[CrossRef]

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006).
[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, 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]

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

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

F. G. Deng, X. H. Li, C. Y. Li, P. Zhou, and H. Y. Zhou, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein–Podolsky–Rosen pairs,” Phys. Rev. A 72, 044301 (2005).
[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]

M. Czechlewski, A. Grudka, S. Ishizaka, and A. Wójcik, “Entanglement purification protocol for a mixture of a pure entangled state and a pure product state,” Phys. Rev. A 80, 014303 (2009).
[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 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]

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

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

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005).
[CrossRef]

Phys. Rev. Lett.

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

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef]

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

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[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]

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]

Quantum Inf. Comput.

C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

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).

Sci. Chin. Ser. G

B. Gu, S. X. Pei, B. Song, and K. Zhong, “Deterministic secure quantum communication over a collective-noise channel,” Sci. Chin. Ser. G 52, 1913–1918 (2009).
[CrossRef]

Other

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1.

Schematic diagram of the proposed ECP for three-photon W states with an ancillary single photon and linear optical elements. PBS, polarization beam splitter; R45, Hadamard operation on the polarization of a single photon; H/V, single-photon measurement with the basis Z={H,V}. When Carson obtains the outcome of measurement on the photon D1 |HD1, the three parties obtain a standard three-photon W state. When he obtains |VD1, Carson need only to operate the photon C1 with the unitary operation σz=|HH||VV| for obtaining the standard three-photon W state.

Equations (13)

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

|ϕABC=α|HA|HB|VC+β(|HA|VB|HC+|VA|HB|HC),
|α|2+2|β|2=1.
|ϕD=1α2+β2(α|HD+β|VD).
|ΨABCD=|ϕABC|ϕD=1α2+β2{α2|HA|HB|VC|HD+β2(|HA|VB+|VA|HB)|HC|VD+αβ[|HA|HB|VC|VD+(|HA|VB+|VA|HB)|HC|HD]}.
|ΨABC1D1=1α2+β2αβ[|HA|HB|VC1|VD1+(|HA|VB+|VA|HB)|HC1|HD1].
|H12(|H+|V),|V12(|H|V).
|ΨABC1D1=16[(|HA|HB|VC1+|HA|VB|HC1+|VA|HB|HC1)|HD1+(|HA|HB|VC1+|HA|VB|HC1+|VA|HB|HC1)|VD1].
P3=3α2β2α2+β2.
|ϕNABCZ=α|VA|HB|HC|HZ+β(|HA|VB|HC|HZ+|HA|HB|VC|HZ++|HA|HB|HC|VZ).
|α|2+(N1)|β|2=1.
|ΨaABCZ=|ϕa|ϕNABCZ=1α2+β2(α|H+β|V)a[α|VA|HB|HC|HZ+β|HA(|VB|HC|HZ+|HB|VC|HZ++|HB|HC|VZ)]=1α2+β2{α2|Ha|VA|HB|HC|HZ+β2|Va|HA(|VB|HC|HZ+|HB|VC|HZ++|HB|HC|VZ)+αβ[|Ha|HA(|VB|HC|HZ+|HB|VC|HZ++|HB|HC|VZ)+|Va|VA|HB|HC|HZ]}.
|ΨaABCZ=αβα2+β2{|Ha|HA(|VB|HC|HZ+|HB|VC|HZ++|HB|HC|VZ)+|Va|VA|HB|HC|HZ}.
PN=Nα2β2α2+β2.

Metrics