Abstract

The entanglement concentration protocols (ECPs) are presented for distilling the maximally entangled state from a known partially entangled n-photon Bell-class and W state, respectively, only resorting to the linear optical elements and quantum nondemolition detector. Different from the traditional parity check with a single photon as an ancilla, we use linear optical elements for changing the photon paths to realize the concentration. The total success probability of the concentration for n-photon entangled states is calculated. The present protocols only require a partially entangled state and do not need any single auxiliary particle. By iterating the ECP repeatedly, the maximal success probability can be increased. Meanwhile, the present protocols are more suitable for the photon system due to the simple operations.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  32. F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012).
    [CrossRef]
  33. B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
    [CrossRef]
  34. S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
    [CrossRef]
  35. C. R. Zhao and L. Ye, “Robust scheme for the preparation of symmetric Dicke states with coherence state via cross-Kerr nonlinearity,” Opt. Commun. 284, 541–544 (2011).
    [CrossRef]
  36. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
    [CrossRef]

2013 (1)

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

2012 (5)

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]

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

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

2011 (2)

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. R. Zhao and L. Ye, “Robust scheme for the preparation of symmetric Dicke states with coherence state via cross-Kerr nonlinearity,” Opt. Commun. 284, 541–544 (2011).
[CrossRef]

2010 (4)

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 0272–0281 (2010).

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]

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]

2008 (2)

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]

2007 (1)

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

2006 (2)

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]

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

2005 (4)

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]

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, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

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

2004 (1)

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

2003 (2)

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

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[CrossRef]

2002 (2)

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

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[CrossRef]

2001 (2)

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)

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

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

1998 (1)

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

1996 (1)

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]

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 (1)

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography using any two nonorthogonal states,” 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]

Bennett, C. H.

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 using any two nonorthogonal states,” Phys. Rev. Lett. 68, 557–559 (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. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[CrossRef]

Bose, S.

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[CrossRef]

Bourennane, M.

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

Brassard, G.

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 using any two nonorthogonal states,” Phys. Rev. Lett. 68, 557–559 (1992).
[CrossRef]

Brune, M.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Buzek, V.

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

Chen, Y. A.

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

Cheng, L. Y.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[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]

Deng, F. G.

F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon 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, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[CrossRef]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 0272–0281 (2010).

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

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]

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, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (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]

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]

Du, F. F.

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bells 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]

Gao, T.

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

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]

Haroche, S.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Hillery, M.

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

Imoto, N.

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[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]

Jin, G. S.

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

Jozsa, R.

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

Koashi, M.

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[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]

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]

Li, C. Y.

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

Li, T.

Li, X. H.

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]

Li, Y.

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.

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, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (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]

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

Long, G. L.

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

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

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

Louis, S. G. R.

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

Man, Z. X.

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 using any two nonorthogonal states,” 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]

Munro, W. J.

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

Nemoto, K.

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

Nogues, G.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Omar, Y.

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[CrossRef]

Osnaghi, S.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Ozdemir, S. K.

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[CrossRef]

Pan, J. W.

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

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

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

Paunkovic, N.

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[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.

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]

Raimond, J. M.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Rauschenbeutel, A.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

Ren, B. C.

Schumacher, B.

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

Sheng, Y. B.

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

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

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 0272–0281 (2010).

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

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]

Si, B.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

Spiller, T. P.

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

Su, S. L.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

Sun, L. L.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

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

Vedral, V.

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[CrossRef]

Wang, C.

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

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

Wang, H. F.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

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

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]

Wang, Y.

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (2005).
[CrossRef]

Wei, H. R.

Wootters, W. K.

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]

Yamamoto, T.

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[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, T.

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

Ye, L.

C. R. Zhao and L. Ye, “Robust scheme for the preparation of symmetric Dicke states with coherence state via cross-Kerr nonlinearity,” Opt. Commun. 284, 541–544 (2011).
[CrossRef]

Yeon, K. H.

Zhan, M. S.

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

Zhang, A. N.

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

Zhang, C. Y.

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, S.

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

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

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]

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

Zhang, Y.

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]

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]

Zhao, C. R.

C. R. Zhao and L. Ye, “Robust scheme for the preparation of symmetric Dicke states with coherence state via cross-Kerr nonlinearity,” Opt. Commun. 284, 541–544 (2011).
[CrossRef]

Zhao, S. M.

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

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

Zhao, Z.

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

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

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 0272–0281 (2010).

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

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]

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (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]

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]

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. B (1)

B. Si, S. L. Su, L. L. Sun, L. Y. Cheng, H. F. Wang, and S. Zhang, “Efficient three-step entanglement concentration for an arbitrary four-photon cluster state,” Chin. Phys. B 22, 030305 (2013).
[CrossRef]

Chin. Phys. Lett. (1)

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]

Eur. Phys. J. D (1)

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

J. Opt. Soc. Am. B (3)

Nature (2)

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999).
[CrossRef]

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421, 343–346 (2003).
[CrossRef]

New J. Phys. (1)

S. G. R. Louis, K. Nemoto, W. J. Munro, and T. P. Spiller, “The efficiencies of generating cluster states with weak nonlinearities,” New J. Phys. 9, 193 (2007).
[CrossRef]

Opt. Commun. (2)

C. R. Zhao and L. Ye, “Robust scheme for the preparation of symmetric Dicke states with coherence state via cross-Kerr nonlinearity,” Opt. Commun. 284, 541–544 (2011).
[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]

Phys. Rev. A (19)

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

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]

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]

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]

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

F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Multiparty quantum-state sharing of an arbitrary two-particle state with Einstein-Podolsky-Rosen pairs,” Phys. Rev. A 72, 022338 (2005).
[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]

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

F. L. Yan and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
[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, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
[CrossRef]

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

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]

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

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

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

Phys. Rev. Lett. (5)

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]

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

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

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[CrossRef]

N. Paunković, Y. Omar, S. Bose, and V. Vedral, “Entanglement concentration using quantum statistics,” Phys. Rev. Lett. 88, 187903 (2002).
[CrossRef]

Quantum Inf. Comput. (1)

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 0272–0281 (2010).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic illustration for the concentration of photon partially entangled Bell-class state. S is the partially entangled source. Alice and Bob share a partially entangled two-photon state. The HWPi (i=a,b) is a half-wave plate and performs the transformations {|Hcos2θi|H+sin2θi|V,|Vcos2θi|V+sin2θi|H} and the same hereinafter.

Fig. 2.
Fig. 2.

Schematic illustration for concentration of three-photon partially entangled W state.

Fig. 3.
Fig. 3.

Schematic illustration for concentration of n-photon partially entangled W state.

Fig. 4.
Fig. 4.

Success probabilities of ECPs for partially entangled Bell-class state (left) and partially entangled three-photon W state (right). For the three-photon W state b=1/3 has been taken for the sake of simplicity, and N(M) denotes the iteration times of Alice (Charlie).

Fig. 5.
Fig. 5.

Success probabilities of the ECPs for partially entangled n-photon (n=4, 5, 6) W state. We have taken ai=(1/n), (i=2,3,,n1) for the sake of simplicity, and i4(5,6) denotes the iteration times of the (n1) partners.

Equations (29)

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

|Hcos2θi|H+sin2θi|V,|Vcos2θi|V+sin2θi|H.
|ΨAB=α|HA|HB+β|VA|VB,
|Ψ1=α(α|H+β|V)1A|HB+β(α|H+β|V)2A|VB,
|Ψ1=α2|H1A|HB+β2|V1A|VB+αβ|H2A|HB+αβ|V2A|VB.
|Ψ2=12(|HA|HB+|VA|VB),
|Ψ3=α2α4+β4|HA|HB+β2α4+β4|VA|VB.
P2=2(α4+β4)(α2α4+β4)2(β2α4+β4)2=2α4β4α4+β4.
PN=2α2β2α2+β2+2α4β4(α2+β2)(α4+β4)++2α2Nβ2N(α2+β2)(α4+β4)(α2N+β2N).
|Ψn=α|HA|HB|H+β|VA|VB|V,
|ΦABC=a|HA|VB|VC+b|VA|HB|VC+c|VA|VB|HC,
|Φ1=acos2θa|H1A|VB|VC+bsin2θa|V1A|HB|VC+csin2θa|V1A|VB|HC+asin2θa|H2A|VB|VC+bcos2θa|V2A|HB|VC+ccos2θa|V2A|VB|HC,
|Φ2=acos2θa|HA|VB|VC+bsin2θa|VA|HB|VC+csin2θa|VA|VB|HC.
|Φ2=aba2+b2(|HA|VB|VC+|VA|HB|VC)+aca2+b2|VA|VB|HC.
|Φ2=bc2+2b2(|HA|VB|VC+|VA|HB|VC)+cc2+2b2|VA|VB|HC.
P11=a2(2b2+c2)a2+b2.
|Φ2=asin2θa|H2A|VB|VC+bcos2θa|V2A|HB|VC+ccos2θa|V2A|VB|HC.
|Φ2=d|HA|VB|VC+d|VA|HB|VC+e|VA|VB|HC.
|Φ3=dsin2θa(|HA|VB|V1C+|VA|HB|V1C)+ecos2θa|VA|VB|H1C+dcos2θa(|HA|VB|V2C+|VA|HB|V2C)+esin2θa|VA|VB|H2C.
|Φ4=dsin2θa(|HA|VB|V1C+|VA|HB|V1C)+ecos2θa|VA|VB|H1C.
|Φ4=13(|HA|VB|VC+|VA|HB|VC+|VA|VB|HC).
P21=3b2c2(b2+c2)(2b2+c2).
|Φ4=dcos2θa(|HA|VB|V2C+|VA|HB|V2C)+esin2θa|VA|VB|H2C.
P1=P11×P21=a2(2b2+c2)a2+b2×3b2c2(b2+c2)(2b2+c2)=3a2b2c2(a2+b2)(b2+c2).
Ptotal=P1N·P2M
P1N=a2N(b2N2c2+2b2N)(a2+b2)(a4+b4)(a2N+b2N),P2M=3b2Mc2M(b2+c2)(b2M+c2M)(2b2+c2),
Wn=1n(|1H|(n1)V+n1|1V|Wn1),
Wn=a1|HVVV+a2|VHVV+a3|VVHV++an|VVVH,
Ptotal=P1i1×P3i3×P4i4××Pnin
P1i1=a22i12a12i1(2a22+a32++an2)(a12+a22)(a14+a24)(a12i1+a22i1),P3i3=a22i32a32i3(3a22+a42++an2)(a22+a32)(a24+a34)(a22i3+a32i3)(2a22+a32++an2),P4i4=a22i42a42i4(4a22+a52++an2)(a22+a42)(a24+a44)(a22i4+a42i4)(3a22+a42++an2),Pkik=a22ik2ak2ik(ka22+ak+12++an2)(a22+ak2)(a24+ak4)(a22ik+ak2ik)[(k1)a22+ak2++an2],Pnin=na22inan2in(a22+an2)(a24+an4)(a22in+an2in)[(n1)a22+an2],

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