Abstract

We present two hyperconcentration protocols for hyperentangled states, in which entanglement occurs simultaneously in the polarization and the spatial degrees of freedom. One uses an auxiliary single photon prepared in a fixed state. The other uses two less-entangled photon pairs. In both schemes, a two photon maximally hyperentangled state can be obtained from the nonmaximally entangled states with a certain success probability. The procrustean concentration is realized by polarizing beam splitters and nondestructive quantum nondemolition detection. In both protocols the unsuccessful instances can be reconcentrated repeatedly to get a higher success probability, which makes our schemes efficient and useful in quantum information processing.

© 2013 Optical Society of America

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2012 (5)

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

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

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

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]

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

2011 (7)

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. Zhu and G. Huang, “Giant Kerr nonlinearity, controlled entangled photons and polarization phase gates in coupled quantum-well structures,” Opt. Express 19, 23364–23376 (2011).
[CrossRef]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (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]

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).

2010 (7)

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

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

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, 0272 (2010).

A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon 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]

2009 (3)

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, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[CrossRef]

G. Vallone, R. Ceccarelli, F. De. Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301(R) (2009).
[CrossRef]

2008 (4)

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

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]

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (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)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

2006 (2)

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]

S. P. Walborn, M. P. Almeida, P. H. Souto Ribeiro, and C. H. Monken, “Quantum information processing with hyperentangled photon states,” Quantum Inf. Comput. 6, 336–350 (2006).

2005 (2)

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

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]

2004 (1)

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

2002 (1)

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

2001 (3)

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]

J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (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 (1)

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

1998 (1)

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[CrossRef]

1997 (1)

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173–2184 (1997).

1996 (3)

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

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

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

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

Almeida, M. P.

S. P. Walborn, M. P. Almeida, P. H. Souto Ribeiro, and C. H. Monken, “Quantum information processing with hyperentangled photon states,” Quantum Inf. Comput. 6, 336–350 (2006).

Barreiro, J. T.

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

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

Bennett, C. H.

C. H. Bennett, 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]

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 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]

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]

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]

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]

Ceccarelli, R.

G. Vallone, R. Ceccarelli, F. De. Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301(R) (2009).
[CrossRef]

Cheng, W. W.

Y. B. Sheng, L. Zhou, W. W. Cheng, L. Y. Gong, and S. M. Zhao, “Efficient electronic entanglement concentration assisted with single mobile electron,” e-print arXiv:quant-ph/1202.2666.

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]

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, “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, 0272 (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, 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, “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]

B. C. Ren, T. J. Wang, M. Hua, F. F. Du, and F. G. Deng, “Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements,” e-print arXiv:quant-ph/1202.2163.

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]

Dowing, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Du, F. F.

B. C. Ren, T. J. Wang, M. Hua, F. F. Du, and F. G. Deng, “Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements,” e-print arXiv:quant-ph/1202.2163.

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]

Feizpour, A.

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Gong, L. Y.

Y. B. Sheng, L. Zhou, W. W. Cheng, L. Y. Gong, and S. M. Zhao, “Efficient electronic entanglement concentration assisted with single mobile electron,” e-print arXiv:quant-ph/1202.2666.

Gu, B.

Guo, G. C.

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

He, B.

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[CrossRef]

Hua, M.

B. C. Ren, T. J. Wang, M. Hua, F. F. Du, and F. G. Deng, “Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements,” e-print arXiv:quant-ph/1202.2163.

Huang, G.

Huang, Y. D.

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[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]

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]

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]

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[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]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Kwiat, P. G.

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

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

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173–2184 (1997).

Langford, N. K.

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

Li, X. H.

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

Lin, Q.

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[CrossRef]

Long, G. L.

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]

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[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]

Martini, F. De.

G. Vallone, R. Ceccarelli, F. De. Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301(R) (2009).
[CrossRef]

Mataloni, P.

G. Vallone, R. Ceccarelli, F. De. Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301(R) (2009).
[CrossRef]

Milbum, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Monken, C. H.

S. P. Walborn, M. P. Almeida, P. H. Souto Ribeiro, and C. H. Monken, “Quantum information processing with hyperentangled photon states,” Quantum Inf. Comput. 6, 336–350 (2006).

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

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

Murao, M.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

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

Nielsen, M. A.

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

Pan, J. W.

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]

Peng, J. D.

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[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]

Peters, N. A.

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

Plenio, M. B.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[CrossRef]

Popescu, S.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[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, 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]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milbum, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Ren, B. C.

B. C. Ren, T. J. Wang, M. Hua, F. F. Du, and F. G. Deng, “Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements,” e-print arXiv:quant-ph/1202.2163.

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, 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 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, “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, 0272 (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, 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, “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]

Y. B. Sheng, L. Zhou, W. W. Cheng, L. Y. Gong, and S. M. Zhao, “Efficient electronic entanglement concentration assisted with single mobile electron,” e-print arXiv:quant-ph/1202.2666.

Y. B. Sheng, L. Zhou, Y. W. Sheng, and S. M. Zhao, “Efficient N-particle W state concentration with different parity check gates,” e-print arXiv:quant-ph/1204.1492.

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient entanglement concentration for three-photon W states with parity check measurement,” e-print arXiv:quant-ph/1202.2616.

Sheng, Y. W.

Y. B. Sheng, L. Zhou, Y. W. Sheng, and S. M. Zhao, “Efficient N-particle W state concentration with different parity check gates,” e-print arXiv:quant-ph/1204.1492.

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.

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[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]

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]

Souto Ribeiro, P. H.

S. P. Walborn, M. P. Almeida, P. H. Souto Ribeiro, and C. H. Monken, “Quantum information processing with hyperentangled photon states,” Quantum Inf. Comput. 6, 336–350 (2006).

Steinberg, A. M.

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Sun, L. L.

Vallone, G.

G. Vallone, R. Ceccarelli, F. De. Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79, 030301(R) (2009).
[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]

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998).
[CrossRef]

Walborn, S. P.

S. P. Walborn, M. P. Almeida, P. H. Souto Ribeiro, and C. H. Monken, “Quantum information processing with hyperentangled photon states,” Quantum Inf. Comput. 6, 336–350 (2006).

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

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[CrossRef]

Wang, H. F.

Wang, T. J.

B. C. Ren, T. J. Wang, M. Hua, F. F. Du, and F. G. Deng, “Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements,” e-print arXiv:quant-ph/1202.2163.

Wei, T. C.

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

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]

Xiao, L.

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[CrossRef]

Xing, X.

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Xiong, W.

Yamamoto, T.

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

Yang, 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.

Yildiz, A.

A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010).
[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, S.

Zhang, W.

L. Xiao, C. Wang, W. Zhang, Y. D. Huang, J. D. Peng, and G. L. Long, “Efficient strategy for sharing entanglement via noisy channels with doubly entangled photon pairs,” Phys. Rev. A 77, 042315 (2008).
[CrossRef]

Zhang, Y.

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]

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]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient entanglement concentration for three-photon W states with parity check measurement,” e-print arXiv:quant-ph/1202.2616.

Y. B. Sheng, L. Zhou, Y. W. Sheng, and S. M. Zhao, “Efficient N-particle W state concentration with different parity check gates,” e-print arXiv:quant-ph/1204.1492.

Y. B. Sheng, L. Zhou, W. W. Cheng, L. Y. Gong, and S. M. Zhao, “Efficient electronic entanglement concentration assisted with single mobile electron,” e-print arXiv:quant-ph/1202.2666.

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.

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

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]

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient entanglement concentration for three-photon W states with parity check measurement,” e-print arXiv:quant-ph/1202.2616.

Y. B. Sheng, L. Zhou, Y. W. Sheng, and S. M. Zhao, “Efficient N-particle W state concentration with different parity check gates,” e-print arXiv:quant-ph/1204.1492.

Y. B. Sheng, L. Zhou, W. W. Cheng, L. Y. Gong, and S. M. Zhao, “Efficient electronic entanglement concentration assisted with single mobile electron,” e-print arXiv:quant-ph/1202.2666.

Zhu, C.

Eur. Phys. J. D (2)

Y. B. Sheng, F. G. Deng, 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 states,” Eur. Phys. J. D 56, 271–275 (2010).
[CrossRef]

J. Mod. Opt. (1)

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173–2184 (1997).

J. Opt. Soc. Am. B (4)

Nat. Phys. (1)

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

Nature (1)

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

Opt. Express (1)

Phys. Lett. A (2)

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

Phys. Rev A (1)

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

Phys. Rev. A (20)

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]

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, “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]

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]

A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010).
[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]

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]

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[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Schematic diagram of the present hyperconcentration protocol. The maximal hyperentanglement is distilled with the help of the auxiliary single-photon state. QND and PBSs are used to project the system into the desired state. The quantum state of photons A and B will collapse to a maximally entangled state with a certain probability.

Fig. 2.
Fig. 2.

Schematic diagram of the proposed hyperconcentration scheme. Two identical less entangled photon pairs are sent to two parties. Alice possesses photons A and C and Bob controls photons B and D. The HWPs flip the horizontal and vertical polarizations. PBSs and QND are used to implement the procrustean concentration. After two photons are measured in the |± basis, the two parties can obtain the maximally hyperentangled state with a certain probability.

Fig. 3.
Fig. 3.

Success probability is altered with the parameters of the initial less-entangled state.

Fig. 4.
Fig. 4.

Schematic diagram of the device that changes the state corresponding to a ±θ phase shift to the less-entangled state to be distilled in the next round.

Fig. 5.
Fig. 5.

Total success probability Ptotal is altered with the coefficient of the initial state and the iteration number of the hyperconcentration process. Here, we chose α0=0.6 and n=0, 1, 2, 3 as an example. Here, n=0 means only the first round of concentration is performed, without iteration.

Fig. 6.
Fig. 6.

Yield Y is altered with the iteration number of hyperconcentration processes and the coefficient δ0. Here, we chose α0=0.6 and n=0, 1, 2, 3 as an example, where n=0 means only the first round of concentration is performed, without iteration.

Equations (36)

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H=χasasapap,
|ψs=c0|0s+c1|1,
U|ψs|αp=c0|0s|αp+c1|1s|αeiθp.
|ψAB=(α0|HH+β0|VV)AB(δ0|a1b1+η0|a2b2)AB.
|α0|2+|β0|2=1,|δ0|2+|η0|2=1.
|ΨMES=12(|HH+|VV)12(|a1b1+|a2b2).
|φX0=(α0|V+β0|H)X0(δ0|x1+η0|x2)X0.
|Ψ=|ψAB|φX0.
|Ψ=α0β0δ02(|HHHa1b2x2+|VVVa1b1x1)+α0β0η02(|HHHa2b1x1+|VVVa2b2x2)+α0β0δ0η0(|HHHa1b1x2+|VVVa1b1x2+|HHHa2b2x1+|VVVa2b2x1).
|Ψ=α0β0δ0η0(|HHH+|VVV)(|a1b1x2+|a2b2x1).
x112(x1+x2);x212(x1x2).
|Ψ=|+x1(|HH+|VV)(|a1b1+|a2b2)+|x1(|HH|VV)(|a1b1+|a2b2)+|+x2(|HH+|VV)(|a1b1|a2b2)+|x2(|HH|VV)(|a1b1|a2b2).
|ψCD=(α0|HH+β0|VV)CD(δ0|c1d1+η0|c2d2)CD.
|ΦABCD=|ψAB|ψCD.
[(α0|HH+β0|VV)(δ0|a1b1+η0|a2b2)]AB[(α0|VV+β0|HH)(δ0|c1d1+η0|c2d2)CD].
|Φ=α0β0δ02(|HHHHa1b2c1d2+|VVVVa1b1c1d1)+α0β0η02(|HHHHa2b1c2d1+|VVVVa2b2c2d2)+α0β0δ0η0(|HHHHa1b1c2d2+|VVVVa1b1c2d2+|HHHHa2b2c1d1+|VVVVa2b2c1d1).
|Φ=|HHHHa1b1c2d2+|VVVVa1b1c2d2+|HHHHa2b2c1d1+|VVVVa2b2c1d1.
|Φ=(|++c1d1+|c1d1+|++c2d2+|c2d2)(|HH+|VV)(|a1b1+|a2b2)+(|+c1d1+|+c1d1+|+c2d2+|+c2d2)(|HH|VV)(|a1b1+|a2b2)(|++c1d2+|c1d2+|++c2d1+|c2d1)(|HH+|VV)(|a1b1|a2b2)(|+c1d2+|+c1d2+|+c2d1+|+c2d1)(|HH|VV)(|a1b1|a2b2).
|Ψ1=α0β0δ02(|HHHa1b2x2+|VVVa1b1x1)+α0β0η02(|HHHa2b1x1+|VVVa2b2x2).
|Ψ1=12δ1(|HHa1b2+|VVa1b1)+12η1(|HHa2b1+|VVa2b2).
|Φ1=α0β0δ02(|HHHHa1b2c1d2+|VVVVa1b1c1d1)+α0β0η02(|HHHHa1b1c2d1+|VVVVa2b2c2d2).
|Φ1=12δ1(|HHa1b2+|VVa1b1)+12η1(|HHa2b1+|VVa2b2).
|Φ1HWP12δ1(|HVa1b2+|VHa1b1)+12η1(|HVa2b1+|VHa2b2)PBS12δ1(|HVa1b1+|VHa1b1)+12η1(|HVa2b2+|VHa2b2).=12(|HV+|VH)(δ1|a1b1+η1|a2b2)HWP12(|HH+|VV)(δ1|a1b1+η1|a2b2).
(α1|HH+β1|VV)(δ1|a1b1+η1|a2b2),
|φX1=(α1|V+β1|H)X1(δ1|x1+η1|x2)X1.
P1=4α12β12δ12η12=δ12η12=δ04η04δ04+η04.
P1=2α12β12(δ14+η14)=12δ08+η08(δ04+η04)2,
(α2|HH+β2|VV)(δ2|a1b1+η2|a2b2),
δk=δk12(δk14+ηk14),
ηk=ηk12(δk14+ηk14).
Pk=4αk2βk2δk2ηk2=(δ0η0)2k+1(δ02k+1+η02k+1)2,
Pk=2αk2βk2(δk4+ηk4)=12δ02k+2+η02k+2(δ02k+1+η02k+1)2.
Pt=P0+P0P1+P0P1P2++P0P1Pn1Pn=4α02β02δ02η02+2α02β02[δ04η04δ04+η04+12δ08η08(δ04+η04)(δ08+η08)++12n1(δ0η0)2n+1(δ04+η04)(δ08+η08)(δ02n+1+η02n+1)].
Yi=NmNl,
Y1=122P0P1=12α02β02δ04η04δ04+η04,Y2=123P0P1P2=14α02β02δ08η08(δ04+η04)(δ08+η08),,Yn=12n+1P0P1P2Pn1Pn=122n1α02β02(δ0η0)2n+1(δ04+η04)(δ08+η08)(δ02n+1+η02n+1)],=122n1α02β02(δ0η0)2n+1j=1n(δ02j+1+η02j+1).
Y=i=0nYi.

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