Abstract

We present two three-photon entanglement concentration protocols (ECPs) for an arbitrary unknown less-entangled W-class state, resorting to linear optical elements. In our normal ECP, the three parties can obtain a three-photon system in a standard W state probabilistically, similar to the Schmidt projection method. In our improved ECP, the parties obtain not only some higher-fidelity partially entangled three-photon systems but also some entangled two-photon systems in each round of concentration with two copies of three-photon systems. It is interesting to show that the three-photon and two-photon systems kept after a round of concentration have the same parameters. The parties can obtain some three-photon systems in a standard W state with a far higher success probability than the normal ECP, by exploiting the three-photon and two-photon systems with the same parameters as the resource for the next round of concentration. Both of these ECPs may have good applications in quantum communication in the future.

© 2013 Optical Society of America

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    [CrossRef]
  29. 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).
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  38. B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000).
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  39. T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  45. 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]
  46. 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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    [CrossRef]
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    [CrossRef]
  51. 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]

2012 (5)

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]

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]

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]

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 (4)

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]

F. G. Deng, X. H. Li, and H. Y. Zhou, “Passively self-errorrejecting qubit transmission over a collective-noise channel,” Quantum Inf. Comput. 11, 0913–0924 (2011).

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

2010 (4)

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

Y. B. Sheng and F. G. Deng, “Efficient quantum entanglement distribution over an arbitrary collective-noise channel,” Phys. Rev. A 81, 042332 (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]

2008 (2)

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 downconversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

2007 (3)

X. H. Li, F. G. Deng, and H. Y. Zhou, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[CrossRef]

Z. X. Man, Y. J. Xia, and N. B. An, “Quantum state sharing of an arbitrary multiqubit state using nonmaximally entangled GHZ states,” Eur. Phys. J. D 42, 333–340 (2007).
[CrossRef]

Z. Y. Wang, H. Yuan, S. H. Shi, and Z. J. Zhang, “Three-party qutrit-state sharing,” Eur. Phys. J. D 41, 371–375 (2007).
[CrossRef]

2006 (3)

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 (7)

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]

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

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 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]

T. Yamamoto, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (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]

2004 (4)

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
[CrossRef]

J. C. Boileau, R. Laflamme, M. Laforest, and C. R. Myers, “Robust quantum communication using a polarization-entangled photon pair,” Phys. Rev. Lett. 93, 220501 (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]

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 (1)

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[CrossRef]

2002 (3)

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

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

2001 (3)

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 (1)

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

1999 (2)

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]

1998 (1)

A. Karlsson and M. Bourennane, “Quantum teleportation using three-particle entanglement,” Phys. Rev. A 58, 4394–4400 (1998).
[CrossRef]

1996 (3)

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 (1)

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 (2)

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, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[CrossRef]

1991 (1)

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

Abouraddy, A. F.

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[CrossRef]

An, N. B.

Z. X. Man, Y. J. Xia, and N. B. An, “Quantum state sharing of an arbitrary multiqubit state using nonmaximally entangled GHZ states,” Eur. Phys. J. D 42, 333–340 (2007).
[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 Bells 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]

Boileau, J. C.

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
[CrossRef]

J. C. Boileau, R. Laflamme, M. Laforest, and C. R. Myers, “Robust quantum communication using a polarization-entangled photon pair,” Phys. Rev. Lett. 93, 220501 (2004).
[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]

Bourennane, M.

A. Karlsson and M. Bourennane, “Quantum teleportation using three-particle entanglement,” Phys. Rev. A 58, 4394–4400 (1998).
[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 Bells 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]

Chuang, I. L.

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

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]

Deng, F. G.

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

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

F. G. Deng, X. H. Li, and H. Y. Zhou, “Passively self-errorrejecting qubit transmission over a collective-noise channel,” Quantum Inf. Comput. 11, 0913–0924 (2011).

Y. B. Sheng and F. G. Deng, “Efficient quantum entanglement distribution over an arbitrary collective-noise channel,” Phys. Rev. A 81, 042332 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric downconversion 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, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[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]

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, 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]

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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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).
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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).
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A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
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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]

Gao, T.

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

Gottesman, D.

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
[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).
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M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
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T. Yamamoto, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (2005).
[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]

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]

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).
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A. Karlsson and M. Bourennane, “Quantum teleportation using three-particle entanglement,” Phys. Rev. A 58, 4394–4400 (1998).
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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, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (2005).
[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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Laflamme, R.

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
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J. C. Boileau, R. Laflamme, M. Laforest, and C. R. Myers, “Robust quantum communication using a polarization-entangled photon pair,” Phys. Rev. Lett. 93, 220501 (2004).
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J. C. Boileau, R. Laflamme, M. Laforest, and C. R. Myers, “Robust quantum communication using a polarization-entangled photon pair,” Phys. Rev. Lett. 93, 220501 (2004).
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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. Y.

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]

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, T.

Li, X. H.

F. G. Deng, X. H. Li, and H. Y. Zhou, “Passively self-errorrejecting qubit transmission over a collective-noise channel,” Quantum Inf. Comput. 11, 0913–0924 (2011).

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, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[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, 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]

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]

Liu, X. S.

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.

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]

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. X. Man, Y. J. Xia, and N. B. An, “Quantum state sharing of an arbitrary multiqubit state using nonmaximally entangled GHZ states,” Eur. Phys. J. D 42, 333–340 (2007).
[CrossRef]

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

Mermin, N. D.

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

Myers, C. R.

J. C. Boileau, R. Laflamme, M. Laforest, and C. R. Myers, “Robust quantum communication using a polarization-entangled photon pair,” Phys. Rev. Lett. 93, 220501 (2004).
[CrossRef]

Nielsen, M. A.

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

Özdemir, S. K.

T. Yamamoto, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (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]

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

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

Poulin, D.

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
[CrossRef]

Ren, B. C.

Saleh, B. E. A.

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[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]

Sergienko, A. V.

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[CrossRef]

Sheng, Y. B.

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]

Y. B. Sheng and F. G. Deng, “Efficient quantum entanglement distribution over an arbitrary collective-noise channel,” Phys. Rev. A 81, 042332 (2010).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization entanglement purification based on parametric downconversion 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]

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]

Shi, S. H.

Z. Y. Wang, H. Yuan, S. H. Shi, and Z. J. Zhang, “Three-party qutrit-state sharing,” Eur. Phys. J. D 41, 371–375 (2007).
[CrossRef]

Shimamura, J.

T. Yamamoto, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (2005).
[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, 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]

Spekkens, R. W.

J. C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, and R. W. Spekkens, “Robust polarization-based quantum key distribution over a collective-noise channel,” Phys. Rev. Lett. 92, 017901 (2004).
[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]

Teich, M. C.

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[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]

Walton, Z. D.

Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Decoherence-free subspaces in quantum key distribution,” Phys. Rev. Lett. 91, 087901 (2003).
[CrossRef]

Wang, C.

Wang, H. F.

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]

Wang, H.-F.

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. Y.

Z. Y. Wang, H. Yuan, S. H. Shi, and Z. J. Zhang, “Three-party qutrit-state sharing,” Eur. Phys. J. D 41, 371–375 (2007).
[CrossRef]

Wei, H. R.

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]

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. J.

Z. X. Man, Y. J. Xia, and N. B. An, “Quantum state sharing of an arbitrary multiqubit state using nonmaximally entangled GHZ states,” Eur. Phys. J. D 42, 333–340 (2007).
[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.

Yamamoto, T.

T. Yamamoto, J. Shimamura, S. K. Özdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (2005).
[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]

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]

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]

Ye, L.

Yeon, K. H.

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]

Yeon, K.-H.

Yuan, H.

Z. Y. Wang, H. Yuan, S. H. Shi, and Z. J. Zhang, “Three-party qutrit-state sharing,” Eur. Phys. J. D 41, 371–375 (2007).
[CrossRef]

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, 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, R.

Zhang, S.

Zhang, Y.

Zhang, Z. J.

Z. Y. Wang, H. Yuan, S. H. Shi, and Z. J. Zhang, “Three-party qutrit-state sharing,” Eur. Phys. J. D 41, 371–375 (2007).
[CrossRef]

Z. J. Zhang, Y. Li, and Z. X. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71, 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]

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).
[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]

Zhou, H. Y.

F. G. Deng, X. H. Li, and H. Y. Zhou, “Passively self-errorrejecting qubit transmission over a collective-noise channel,” Quantum Inf. Comput. 11, 0913–0924 (2011).

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 downconversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

X. H. Li, F. G. Deng, and H. Y. Zhou, “Faithful qubit transmission against collective noise without ancillary qubits,” Appl. Phys. Lett. 91, 144101 (2007).
[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]

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Zhou, P.

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Quantum Inf. Comput. (1)

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Figures (5)

Fig. 1.
Fig. 1.

Principle of our normal ECP for obtaining a standard W state from three-photon unknown W -class states with linear optical elements. A i B i C i ( i = 1 , 2 , 3 ) is a three-photon system in a partially entangled W state. PBS represents a polarizing beam splitter, which transmits the photon in the horizontal polarization state | H and reflects the photon in the vertical polarization state | V . HWP represents a half-wave plate that acts as a Hadamard operation on the polarizations of photons. Q H and Q V ( Q = A , B , C ) represent two single-photon detectors for distinguishing the polarization states | H and | V , respectively.

Fig. 2.
Fig. 2.

Principle of the first step in our improved ECP for obtaining a standard W state from three-photon unknown W -class states with linear optical elements. HWP represents a Hadamard operation on the polarizations of photons. BS is a 50 50 beam splitter. PBS represents a polarizing beam splitter. APNS represents an approximative photon-number splitter composed of some linear optical elements. The discarded instance corresponds to the fact that there is no photon clicking the single-photon detectors placed at the upper path. When there are one or two photons passing through the upper path, the six-photon system is in the state | Ψ 1 C 1 B 1 A 1 C 2 B 2 A 2 or | Ψ 2 C 1 B 1 A 1 C 2 B 2 A 2 , respectively.

Fig. 3.
Fig. 3.

Principle of the improved ECP for obtaining a standard W state from three-photon unknown W -class states with linear optical elements. PBS represents a polarizing beam splitter. HWP represents a Hadamard operation on the polarizations of photons. H and V represent the horizontal polarization states of photons | H and the vertical polarization state | V , respectively.

Fig. 4.
Fig. 4.

Time-divisioned multiplexing for splitting the photons into 2 N wavepackets with N BSs, similar to that in [26].

Fig. 5.
Fig. 5.

Success probabilities P that the three parties distil a three-photon system in a standard W state from N -group sufficient partially entangled resources. The solid line shows the success probability in the normal ECP, and the dotted line shows that in our improved ECP. (a) Relation between the success probability P and the parameter | b | 2 when P 1 = 2 P 2 . (b) Relation between the success probability P and the parameter E = 3 | b | 2 when | a | > | b | and | c | 2 = 1 / 3 .

Equations (29)

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| φ C B A = a | H C | H B | V A + b | H C | V B | H A + c | V C | H B | H A ,
| a | 2 + | b | 2 + | c | 2 = 1 .
| φ C 1 B 1 A 1 = a | H C 1 | H B 1 | H A 1 + b | H C 1 | V B 1 | V A 1 + c | V C 1 | H B 1 | V A 1 , | φ C 2 B 2 A 2 = a | H C 2 | H B 2 | V A 2 + b | H C 2 | V B 2 | H A 2 + c | V C 2 | H B 2 | H A 2 .
| Φ C 1 B 1 A 1 C 2 B 2 A 2 = | φ C 1 B 1 A 1 | φ C 2 B 2 A 2 = a [ | H H C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | H H A 1 A 2 + ( b | H V + c | V H ) C 1 B 1 | H H C 2 B 2 | V V A 1 A 2 ] + a 2 | H H C 1 B 1 | H H C 2 B 2 | H V A 1 A 2 + ( b | H V + c | V H ) C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | V H A 1 A 2 .
| Ψ 1 C 1 B 1 A 1 C 2 B 2 A 2 = a | H H C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | H H A 1 A 2 + a ( b | H V + c | V H ) C 1 B 1 | H H C 2 B 2 | V V A 1 A 2 .
| Ψ 1 C 1 B 1 A 1 C 2 B 2 A 2 = a 2 | H H C 1 B 1 | H H C 2 B 2 | H V A 1 A 2 + ( b | H V + c | V H ) C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | V H A 1 A 2 .
| Φ 1 C 2 B 2 A 2 B 1 A 1 = a ( b | H V H + c | V H H + b | H H V ) C 2 B 2 A 2 ( | H H + | V V ) B 1 A 1 + a ( b | H V H + c | V H H b | H H V ) C 2 B 2 A 2 ( | H V + | V H ) B 1 A 1 .
| φ 1 C 2 B 2 A 2 = α ( b | H V H + b | H H V + c | V H H ) C 2 B 2 A 2 , | φ 1 C 2 B 2 A 2 = α ( b | H V H b | H H V + c | V H H ) C 2 B 2 A 2 .
| φ C 3 B 3 A 3 = a | V C 3 | H B 3 | V A 3 + b | V C 3 | V B 3 | H A 3 + c | H C 3 | H B 3 | H A 3 .
| χ C 3 B 3 A 3 C 2 B 2 A 2 = | φ C 3 B 3 A 3 | φ 1 C 2 B 2 A 2 = α c b | V V C 3 C 2 | V H B 3 A 3 | H H B 2 A 2 + α c b | H H C 3 C 2 | H H B 3 A 3 ( | V H + | H V ) B 2 A 2 + α a c | V V C 3 C 2 | H V B 3 A 3 | H H B 2 A 2 + α a b | V H C 3 C 2 | H V B 3 A 3 ( | V H + | H V ) B 2 A 2 + α b 2 | V H C 3 C 2 | V H B 3 A 3 ( | V H + | H V ) B 2 A 2 + α c 2 | H V C 3 C 2 | H H B 3 A 3 | H H B 2 A 2 .
| χ 1 C 3 B 3 A 3 C 2 B 2 A 2 = α c b | V V C 3 C 2 | V H B 3 A 3 | H H B 2 A 2 + α c b | H H C 3 C 2 | H H B 3 A 3 ( | V H + | H V ) B 2 A 2 + α a c | V V C 3 C 2 | H V B 3 A 3 | H H B 2 A 2 .
| W + C 2 B 2 A 2 = 1 3 ( | H H V + | H V H + | V H H ) C 2 B 2 A 2 .
| W C 2 B 2 A 2 = 1 3 ( | H H V + | H V H | V H H ) C 3 B 3 A 3 .
P s n = 3 | a b c | 2 .
| Ψ 1 C 1 B 1 A 1 C 2 B 2 A 2 = a | H A 1 [ | H H C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | H A 2 + ( b | H V + c | V H ) C 1 B 1 | H H C 2 B 2 | V A 2 ] + a | V A 1 [ | H H C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | H A 2 ( b | H V + c | V H ) C 1 B 1 | H H C 2 B 2 | V A 2 ] .
| Ψ 2 C 1 B 1 A 1 C 2 B 2 A 2 = ( b | H V + c | V H ) C 1 B 1 ( b | H V + c | V H ) C 2 B 2 | V H A 1 A 2 .
| τ 1 C 1 B 1 = β ( b | H V + c | V H ) C 1 B 1
| τ 2 C 2 B 2 = β ( b | H V + c | V H ) C 2 B 2 .
| Ψ 3 C 1 B 1 C 2 B 2 A 1 A 2 = | H H C 1 B 1 | H H C 2 B 2 | H V A 1 A 2 .
| ε = { [ 1 D ^ ( Δ t 1 ) D ^ ( Δ t 2 ) D ^ ( Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 ) D ^ ( Δ t 3 + Δ t 2 ) + D ^ ( Δ t 1 + Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 + Δ t 3 ) ] | a 1 + i [ 1 D ^ ( Δ t 1 ) D ^ ( Δ t 2 ) + D ^ ( Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 ) + D ^ ( Δ t 3 + Δ t 2 ) D ^ ( Δ t 1 + Δ t 3 ) + D ^ ( Δ t 1 + Δ t 2 + Δ t 3 ) ] | a 2 } A 1 { [ 1 D ^ ( Δ t 1 ) D ^ ( Δ t 2 ) D ^ ( Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 ) D ^ ( Δ t 3 + Δ t 2 ) + D ^ ( Δ t 1 + Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 + Δ t 3 ) ] | a 1 + i [ 1 D ^ ( Δ t 1 ) D ^ ( Δ t 2 ) + D ^ ( Δ t 3 ) D ^ ( Δ t 1 + Δ t 2 ) + D ^ ( Δ t 3 + Δ t 2 ) D ^ ( Δ t 1 + Δ t 3 ) + D ^ ( Δ t 1 + Δ t 2 + Δ t 3 ) ] | a 2 } A 2 ( | H H | V V + | H V | V H ) A 1 A 2 .
| ε = [ D ^ ( 0 ) D ^ ( 0 ) + D ^ ( Δ t ) D ^ ( Δ t ) + D ^ ( 2 Δ t ) D ^ ( 2 Δ t ) + D ^ ( 4 Δ t ) D ^ ( Δ 4 t ) + D ^ ( 3 Δ t ) D ^ ( 3 Δ t ) + D ^ ( 6 Δ t ) D ^ ( 6 Δ t ) + D ^ ( 5 Δ t ) D ^ ( 5 Δ t ) + D ^ ( 7 Δ t ) D ^ ( 7 Δ t ) ] A 1 A 2 ( | a 1 a 1 | a 2 a 2 ) A 1 A 2 ( | H H | V V ) A 1 A 2 .
| ω C C B A B = 1 3 [ | H C | H C ( | H | V + | V | H ) B A | H B + | V C | V C | H B | H A | V B ] .
| ω C C B A B = β b 2 | H C | V C ( | H | V + | V | H ) B A | V B + β c 2 | V C | V C | H B | H A | V B .
P r = 3 b 2 c 2 2 b 4 + c 4 + 3 b 2 c 2 .
P r = c 4 + 2 b 4 2 b 4 + c 4 + 3 b 2 c 2 .
| W + C B A = 1 3 ( | H H V + | H V H + | V H H ) C B A
| W C B A = 1 3 ( | H H V + | H V H | V H H ) C B A .
P = ζ 3 | b | 2 | c | 2 2 | b | 4 + | c | 4 + 3 | b | 2 | c | 2 .
P t = P 1 P r = a 2 ( 2 b 2 + c 2 ) 3 b 2 c 2 2 b 4 + c 4 + 3 b 2 c 2 .

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