Abstract

We propose a compact scheme for the heralded generation of single-photon multipartite entanglement by using a single two-dimensional nonlinear photonic crystal. Studies have shown that by appropriate structure design, the single-photon entanglement shared among three spatially distinct optical modes can be generated through three concurrent spontaneous parametric down-conversion processes by using the other photon in an identical spatial mode as a trigger. Furthermore, we analyze the entanglement of such heralded single-photon tripartite W-type state theoretically. This method can be expanded for the heralded single-photon N-partite entanglement generation. This compact and stable quantum light source may act as a key ingredient in quantum information science.

© 2013 OSA

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  24. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
    [CrossRef]
  25. 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(5), 722–725 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  28. N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
    [CrossRef]
  29. S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
    [CrossRef]

2012

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012).
[CrossRef]

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

2011

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express19(25), 25685–25695 (2011).
[CrossRef] [PubMed]

2010

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

2009

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. Y. Lan, A. G. Radnaev, O. A. Collins, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “A multiplexed quantum memory,” Opt. Express17(16), 13639–13645 (2009).
[CrossRef] [PubMed]

2008

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

2007

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

2005

S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A72(6), 064306 (2005).
[CrossRef]

2004

2002

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

2001

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

2000

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
[CrossRef]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000).
[CrossRef]

1998

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998).
[CrossRef]

1997

M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997).
[CrossRef]

1996

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(5), 722–725 (1996).
[CrossRef] [PubMed]

1994

L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett.73(17), 2279–2283 (1994).
[CrossRef] [PubMed]

1991

S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991).
[CrossRef] [PubMed]

Afzelius, M.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

Alexandrescu, A.

Andersen, U. L.

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012).
[CrossRef]

Bai, Y. F.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[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(5), 722–725 (1996).
[CrossRef] [PubMed]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998).
[CrossRef]

Björk, G.

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012).
[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(5), 722–725 (1996).
[CrossRef] [PubMed]

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

Brukner, C.

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

Bussières, F.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

Carrasco, S.

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Choi, K. S.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

Chou, C. W.

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

Cirac, J. I.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000).
[CrossRef]

Clausen, C.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

Coffman, V.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
[CrossRef]

Collett, M. J.

S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991).
[CrossRef] [PubMed]

Collins, O. A.

Coudreau, T.

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

De Martini, F.

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

de Riedmatten, H.

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

Deng, H.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

Duan, L. M.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

Dür, W.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000).
[CrossRef]

Felinto, D.

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

Gisin, N.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

Gong, Y. X.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Guo, G. C.

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

Hardy, L.

L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett.73(17), 2279–2283 (1994).
[CrossRef] [PubMed]

Herrmann, H.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Horne, M. A.

M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997).
[CrossRef]

Jin, H.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Kennedy, T. A. B.

Kimble, H. J.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

Kundu, J.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
[CrossRef]

Kuzmich, A.

Laghaout, A.

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012).
[CrossRef]

Lan, S. Y.

Landry, O.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Laurat, J.

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

Leng, H. Y.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Lombardi, E.

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

Lougovski, P.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

Lu, C. Y.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Lukin, M. D.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

Matsukevich, D. N.

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

Pan, J. W.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

Papp, S. B.

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

Popescu, S.

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

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(5), 722–725 (1996).
[CrossRef] [PubMed]

Radnaev, A. G.

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

Salart, D.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Sangouard, N.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

Sanguinetti, B.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

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(5), 722–725 (1996).
[CrossRef] [PubMed]

Sciarrino, F.

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

Shi, B. S.

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

Simon, C.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

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(5), 722–725 (1996).
[CrossRef] [PubMed]

Sohler, W.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Tan, S. M.

S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991).
[CrossRef] [PubMed]

Thew, R. T.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Thomas, A.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

Torner, L.

Torres, J. P.

Usmani, I.

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

van Enk, S. J.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A72(6), 064306 (2005).
[CrossRef]

Vidal, G.

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000).
[CrossRef]

Walls, D. F.

S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991).
[CrossRef] [PubMed]

Wang, C.

Weinfurter, H.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Wootters, W. K.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
[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(5), 722–725 (1996).
[CrossRef] [PubMed]

Xie, Z. D.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Xu, P.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Yang, J.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

Yang, S.

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

Yu, X. Q.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Zbinden, H.

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

Z-dotukowski, M.

M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997).
[CrossRef]

Zeilinger, A.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997).
[CrossRef]

Zhang, C.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Zhang, R.

Zhang, S. L.

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

Zhang, Y.

Zhu, S. N.

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Zoller, P.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

Zou, X. B.

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

Zukowski, M.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Nat Commun

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011).
[CrossRef] [PubMed]

Nat. Photonics

I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012).
[CrossRef]

Nature

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001).
[CrossRef] [PubMed]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008).
[CrossRef] [PubMed]

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001).
[CrossRef] [PubMed]

New J. Phys.

P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012).
[CrossRef]

N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008).
[CrossRef]

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012).
[CrossRef]

M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997).
[CrossRef]

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000).
[CrossRef]

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000).
[CrossRef]

G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012).
[CrossRef]

S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A72(6), 064306 (2005).
[CrossRef]

Phys. Rev. Lett.

S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991).
[CrossRef] [PubMed]

L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett.73(17), 2279–2283 (1994).
[CrossRef] [PubMed]

D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010).
[CrossRef] [PubMed]

E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002).
[CrossRef] [PubMed]

J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007).
[CrossRef] [PubMed]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998).
[CrossRef]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000).
[CrossRef] [PubMed]

C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007).
[CrossRef] [PubMed]

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(5), 722–725 (1996).
[CrossRef] [PubMed]

Rev. Mod. Phys.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012).
[CrossRef]

Science

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009).
[CrossRef] [PubMed]

C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a) Sketch of a hexagonal poled 2D NPC. (b) Reciprocal lattice of the crystal and the geometries of three concurrent QPM SPDC processes. (c) Transverse pattern of the parametric light in the Fourier plane.

Fig. 2
Fig. 2

(a) The concurrence Ca,b(c) varies with the duty cycle of 2D PPLN crystal. (b) The concurrence Ca,bc varies with the duty cycle of 2D PPLN crystal.

Fig. 3
Fig. 3

Schematic for single-photon tripartite W state generation and entanglement analysis. IF, interference filter; DM, dichroic mirror. Beam-splitter setup to project onto the three |Wj> states: 3 input modes are converted into 3 output modes by 3 lossless beamsplitters. The lowermost beam splitter has a reflectivity R = 1/3, for the other two R = 1/2.

Fig. 4
Fig. 4

(a) Transverse pattern of the parametric light in the Fourier plane. (b) Reciprocal lattice of the crystal and the geometries of four concurrent QPM SPDC processes.

Equations (7)

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χ (2) ( r )= d eff m,n f m,n e i G m,n r ,
H ^ I (t)= ε 0 V d 3 r χ (2) ( r ) E p (+) ( r ,t) E ^ s () ( r ,t) E ^ i () ( r ,t)+H.c.,
|Φ= A 1 dν Ψ 1 (ν) | ω s 1 ω i 1 ab + A 2 dν Ψ 2 (ν) | ω s 2 ω i 2 bb + A 3 dν Ψ 3 (ν) | ω s 3 ω i 3 bc ,
Ψ j (ν)= 1 e iΔ k z j (ν)L iΔ k z j (ν)L = e iΔ k z j (ν)L 2 sinc( Δ k z j (ν)L 2 ),
|Φ= A 1 (| Ω s,a Ω i,b +| Ω s,b Ω i,c )+ A 2 | Ω s,b Ω i,b .
|Φ abc = A 1 η 1 ( | 100 abc + | 001 abc )+ A 2 η 2 | 010 abc ,
Δρ= j=1 3 { Tr(ρ [| W j W j |] 2 ) (Trρ[| W j W j |]) 2 } .

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