Abstract

Cluster state is an important resource for one-way quantum computation and quantum network. In this paper, we present a scheme for connecting two Gaussian cluster states by entanglement swapping, which can be used to connect two local quantum networks composed by cluster states. The connection schemes between different types of four-mode cluster states are analyzed and we show that the structure of the output states after entanglement swapping may be not the same as that of the input states. The entanglement of the obtained new cluster states are presented when suitable feedforward schemes are applied in the entanglement swapping process. By using optimal gains in the classical channel and inseparability criteria, the requirement of squeezing parameters for obtaining entanglement of output states are reduced. The presented scheme provides a concrete reference for constructing quantum networks with cluster states.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  28. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
    [Crossref]
  29. F. Sciarrino, E. Lombardi, G. Milani, and F. De Martini, “Delayed-choice entanglement swapping with vacuum–one-photon quantum states,” Phys. Rev. A 66, 024309 (2002).
    [Crossref]
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    [Crossref]
  31. Y.-B. Sheng, F.-G. Deng, and G. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
    [Crossref]
  32. L. Zhou and Y.-B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
    [Crossref]
  33. R. E. S. Polkinghorne and T. C. Ralph, “Continuous variable entanglement swapping,” Phys. Rev. Lett. 83, 2095-2099 (1999).
    [Crossref]
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    [Crossref]
  35. X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, and K. Peng, “Experimental demonstration of unconditional entanglement swapping for continuous variables,” Phys. Rev. Lett. 93, 250503 (2004).
    [Crossref]
  36. N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
    [Crossref] [PubMed]
  37. S. Takeda, M. Fuwa, P. van Loock, and A. Furusawa, “Entanglement swapping between discrete and continuous variables,” Phys. Rev. Lett. 114, 100501 (2015).
    [Crossref] [PubMed]
  38. U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information,” Nat. Phys. 11, 713-719 (2015).
    [Crossref]
  39. C.-Y. Lu, T. Yang, and J.-W. Pan, “Experimental multiparticle entanglement swapping for quantum networking,” Phys. Rev. Lett. 103, 020501 (2009).
    [Crossref] [PubMed]
  40. X. Su, C. Tian, X. Deng, Q. Li, C. Xie, and K. Peng, “Quantum entanglement swapping between two multipartite entangled states,” Phys. Rev. Lett. 117, 240503 (2016).
    [Crossref] [PubMed]
  41. M. Wang, Z. Qin, and X. Su, “Swapping of Gaussian Einstein-Podolsky-Rosen steering,” Phys. Rev. A 95, 052311 (2017).
    [Crossref]
  42. M. Wang, Z. Qin, Y. Wang, and X. Su, “Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states,” Phys. Rev. A 96, 022307 (2017).
    [Crossref]
  43. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
    [Crossref]

2017 (3)

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

M. Wang, Z. Qin, and X. Su, “Swapping of Gaussian Einstein-Podolsky-Rosen steering,” Phys. Rev. A 95, 052311 (2017).
[Crossref]

M. Wang, Z. Qin, Y. Wang, and X. Su, “Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states,” Phys. Rev. A 96, 022307 (2017).
[Crossref]

2016 (1)

X. Su, C. Tian, X. Deng, Q. Li, C. Xie, and K. Peng, “Quantum entanglement swapping between two multipartite entangled states,” Phys. Rev. Lett. 117, 240503 (2016).
[Crossref] [PubMed]

2015 (3)

S. Takeda, M. Fuwa, P. van Loock, and A. Furusawa, “Entanglement swapping between discrete and continuous variables,” Phys. Rev. Lett. 114, 100501 (2015).
[Crossref] [PubMed]

U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information,” Nat. Phys. 11, 713-719 (2015).
[Crossref]

L. Zhou and Y.-B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
[Crossref]

2014 (2)

P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10, 582-587 (2014).
[Crossref]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

2013 (4)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
[Crossref]

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photon. 8, 109-112 (2013).
[Crossref]

H.-K. Lau and C. Weedbrook, “Quantum secret sharing with continuous-variable cluster states,” Phys. Rev. A 88, 042313 (2013).
[Crossref]

2012 (1)

2011 (2)

N. C. Menicucci, S. T. Flammia, and P. van Loock, “Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices,” Phys. Rev. A 83, 042335 (2011).
[Crossref]

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. I. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref] [PubMed]

2010 (2)

Y.-B. Sheng, F.-G. Deng, and G. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[Crossref]

Y. Miwa, R. Ukai, J.-I. Yoshikawa, R. Filip, P. van Loock, and A. Furusawa, “Demonstration of cluster-state shaping and quantum erasure for continuous variables,” Phys. Rev. A 82, 032305 (2010).
[Crossref]

2009 (3)

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

H. Shen, X. Su, X. Jia, and C. Xie, “Quantum communication network utilizing quadripartite entangled states of optical field,” Phys. Rev. A 80, 042320 (2009).
[Crossref]

C.-Y. Lu, T. Yang, and J.-W. Pan, “Experimental multiparticle entanglement swapping for quantum networking,” Phys. Rev. Lett. 103, 020501 (2009).
[Crossref] [PubMed]

2007 (2)

P. van Loock, “Examples of Gaussian cluster computation,” J. Opt. Soc. Am. B. 24, 340–346 (2007).
[Crossref]

P. van Loock, C. Woodbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A 76, 032321 (2007).
[Crossref]

2006 (2)

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,”, Phys. Rev. Lett. 97110501 (2006).
[Crossref] [PubMed]

2005 (4)

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantum computation,” Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, “Long-distance entanglement swapping with photons from separated sources,” Phys. Rev. A 71, 050302 (2005).
[Crossref]

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[Crossref] [PubMed]

2004 (2)

X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, and K. Peng, “Experimental demonstration of unconditional entanglement swapping for continuous variables,” Phys. Rev. Lett. 93, 250503 (2004).
[Crossref]

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continues variables,” Nature (London) 431, 430-433 (2004).
[Crossref]

2003 (2)

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

2002 (1)

F. Sciarrino, E. Lombardi, G. Milani, and F. De Martini, “Delayed-choice entanglement swapping with vacuum–one-photon quantum states,” Phys. Rev. A 66, 024309 (2002).
[Crossref]

2001 (2)

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[Crossref] [PubMed]

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910-913 (2001).
[Crossref] [PubMed]

2000 (1)

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482-3485 (2000).
[Crossref] [PubMed]

1999 (3)

R. E. S. Polkinghorne and T. C. Ralph, “Continuous variable entanglement swapping,” Phys. Rev. Lett. 83, 2095-2099 (1999).
[Crossref]

S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752-2758 (1999).
[Crossref]

P. van Loock and S. L. Braunstein, “Unconditional teleportation of continuous-variable entanglement,” Phys. Rev. A 61, 010302 (1999).
[Crossref]

1998 (2)

S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822-829 (1998).
[Crossref]

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
[Crossref]

1993 (1)

M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287-4290 (1993).
[Crossref]

Adesso, G.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

Andersen, U. L.

U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information,” Nat. Phys. 11, 713-719 (2015).
[Crossref]

Aoki, T.

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[Crossref] [PubMed]

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continues variables,” Nature (London) 431, 430-433 (2004).
[Crossref]

Armstrong, S. C.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
[Crossref]

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. I. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref] [PubMed]

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Bishof, M.

P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10, 582-587 (2014).
[Crossref]

Bose, S.

S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822-829 (1998).
[Crossref]

Bouwmeester, D.

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
[Crossref]

Braunstein, S. L.

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482-3485 (2000).
[Crossref] [PubMed]

P. van Loock and S. L. Braunstein, “Unconditional teleportation of continuous-variable entanglement,” Phys. Rev. A 61, 010302 (1999).
[Crossref]

Briegel, H. J.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910-913 (2001).
[Crossref] [PubMed]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[Crossref] [PubMed]

Browne, D. E.

D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantum computation,” Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

Chen, M.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

de Araújo, R. M.

J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photon. 8, 109-112 (2013).
[Crossref]

de Riedmatten, H.

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, “Long-distance entanglement swapping with photons from separated sources,” Phys. Rev. A 71, 050302 (2005).
[Crossref]

Deng, F.-G.

Y.-B. Sheng, F.-G. Deng, and G. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[Crossref]

Deng, X.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

X. Su, C. Tian, X. Deng, Q. Li, C. Xie, and K. Peng, “Quantum entanglement swapping between two multipartite entangled states,” Phys. Rev. Lett. 117, 240503 (2016).
[Crossref] [PubMed]

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

Ekert, A. K.

M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287-4290 (1993).
[Crossref]

Fabre, C.

J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photon. 8, 109-112 (2013).
[Crossref]

Filip, R.

Y. Miwa, R. Ukai, J.-I. Yoshikawa, R. Filip, P. van Loock, and A. Furusawa, “Demonstration of cluster-state shaping and quantum erasure for continuous variables,” Phys. Rev. A 82, 032305 (2010).
[Crossref]

Flammia, S. T.

N. C. Menicucci, S. T. Flammia, and P. van Loock, “Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices,” Phys. Rev. A 83, 042335 (2011).
[Crossref]

Furusawa, A.

U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information,” Nat. Phys. 11, 713-719 (2015).
[Crossref]

S. Takeda, M. Fuwa, P. van Loock, and A. Furusawa, “Entanglement swapping between discrete and continuous variables,” Phys. Rev. Lett. 114, 100501 (2015).
[Crossref] [PubMed]

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
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[Crossref]

Vedral, V.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
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S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822-829 (1998).
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Walther, P.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Wang, M.

M. Wang, Z. Qin, Y. Wang, and X. Su, “Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states,” Phys. Rev. A 96, 022307 (2017).
[Crossref]

M. Wang, Z. Qin, and X. Su, “Swapping of Gaussian Einstein-Podolsky-Rosen steering,” Phys. Rev. A 95, 052311 (2017).
[Crossref]

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

Wang, Y.

M. Wang, Z. Qin, Y. Wang, and X. Su, “Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states,” Phys. Rev. A 96, 022307 (2017).
[Crossref]

Weedbrook, C.

H.-K. Lau and C. Weedbrook, “Quantum secret sharing with continuous-variable cluster states,” Phys. Rev. A 88, 042313 (2013).
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M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
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N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,”, Phys. Rev. Lett. 97110501 (2006).
[Crossref] [PubMed]

Weinfurter, H.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
[Crossref]

Woodbrook, C.

P. van Loock, C. Woodbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A 76, 032321 (2007).
[Crossref]

Xiang, Y.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

Xie, C.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

X. Su, C. Tian, X. Deng, Q. Li, C. Xie, and K. Peng, “Quantum entanglement swapping between two multipartite entangled states,” Phys. Rev. Lett. 117, 240503 (2016).
[Crossref] [PubMed]

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

X. Su, Y. Zhao, S. Hao, X. Jia, X. Jia, C. Xie, and K. Peng, “Experimental preparation of eight-partite cluster state for photonic qumodes,” Opt. Lett. 37, 5178-5180 (2012).
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H. Shen, X. Su, X. Jia, and C. Xie, “Quantum communication network utilizing quadripartite entangled states of optical field,” Phys. Rev. A 80, 042320 (2009).
[Crossref]

X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, and K. Peng, “Experimental demonstration of unconditional entanglement swapping for continuous variables,” Phys. Rev. Lett. 93, 250503 (2004).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Yan, Y.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Yang, T.

C.-Y. Lu, T. Yang, and J.-W. Pan, “Experimental multiparticle entanglement swapping for quantum networking,” Phys. Rev. Lett. 103, 020501 (2009).
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Ye, J.

P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10, 582-587 (2014).
[Crossref]

Yokoyama, S.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
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Yonezawa, H.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
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N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
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H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continues variables,” Nature (London) 431, 430-433 (2004).
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Yoshikawa, J. I.

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. I. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
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Yoshikawa, J.-i.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
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Y. Miwa, R. Ukai, J.-I. Yoshikawa, R. Filip, P. van Loock, and A. Furusawa, “Demonstration of cluster-state shaping and quantum erasure for continuous variables,” Phys. Rev. A 82, 032305 (2010).
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Zbinden, H.

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, “Long-distance entanglement swapping with photons from separated sources,” Phys. Rev. A 71, 050302 (2005).
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Zeilinger, A.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
[Crossref]

M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287-4290 (1993).
[Crossref]

Zhang, J.

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Zhao, F.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Zhao, Y.

Zhou, L.

L. Zhou and Y.-B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
[Crossref]

Zukowski, M.

M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287-4290 (1993).
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J. Opt. Soc. Am. B. (1)

P. van Loock, “Examples of Gaussian cluster computation,” J. Opt. Soc. Am. B. 24, 340–346 (2007).
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Nat. Commun. (1)

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

Nat. Photon. (1)

J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photon. 8, 109-112 (2013).
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Nat. Phys. (2)

P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” Nat. Phys. 10, 582-587 (2014).
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U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information,” Nat. Phys. 11, 713-719 (2015).
[Crossref]

Nature (1)

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[Crossref] [PubMed]

Nature (London) (1)

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continues variables,” Nature (London) 431, 430-433 (2004).
[Crossref]

Nature Photon. (1)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nature Photon. 7, 982-986 (2013).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (17)

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006).
[Crossref]

P. van Loock, C. Woodbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A 76, 032321 (2007).
[Crossref]

N. C. Menicucci, S. T. Flammia, and P. van Loock, “Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices,” Phys. Rev. A 83, 042335 (2011).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

H. Shen, X. Su, X. Jia, and C. Xie, “Quantum communication network utilizing quadripartite entangled states of optical field,” Phys. Rev. A 80, 042320 (2009).
[Crossref]

H.-K. Lau and C. Weedbrook, “Quantum secret sharing with continuous-variable cluster states,” Phys. Rev. A 88, 042313 (2013).
[Crossref]

P. van Loock and S. L. Braunstein, “Unconditional teleportation of continuous-variable entanglement,” Phys. Rev. A 61, 010302 (1999).
[Crossref]

Y. Miwa, R. Ukai, J.-I. Yoshikawa, R. Filip, P. van Loock, and A. Furusawa, “Demonstration of cluster-state shaping and quantum erasure for continuous variables,” Phys. Rev. A 82, 032305 (2010).
[Crossref]

S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822-829 (1998).
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F. Sciarrino, E. Lombardi, G. Milani, and F. De Martini, “Delayed-choice entanglement swapping with vacuum–one-photon quantum states,” Phys. Rev. A 66, 024309 (2002).
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H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, “Long-distance entanglement swapping with photons from separated sources,” Phys. Rev. A 71, 050302 (2005).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and G. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
[Crossref]

L. Zhou and Y.-B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A 92, 042314 (2015).
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S. M. Tan, “Confirming entanglement in continuous variable quantum teleportation,” Phys. Rev. A 60, 2752-2758 (1999).
[Crossref]

M. Wang, Z. Qin, and X. Su, “Swapping of Gaussian Einstein-Podolsky-Rosen steering,” Phys. Rev. A 95, 052311 (2017).
[Crossref]

M. Wang, Z. Qin, Y. Wang, and X. Su, “Einstein-Podolsky-Rosen-steering swapping between two Gaussian multipartite entangled states,” Phys. Rev. A 96, 022307 (2017).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

Phys. Rev. Lett. (17)

C.-Y. Lu, T. Yang, and J.-W. Pan, “Experimental multiparticle entanglement swapping for quantum networking,” Phys. Rev. Lett. 103, 020501 (2009).
[Crossref] [PubMed]

X. Su, C. Tian, X. Deng, Q. Li, C. Xie, and K. Peng, “Quantum entanglement swapping between two multipartite entangled states,” Phys. Rev. Lett. 117, 240503 (2016).
[Crossref] [PubMed]

X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, and K. Peng, “Experimental demonstration of unconditional entanglement swapping for continuous variables,” Phys. Rev. Lett. 93, 250503 (2004).
[Crossref]

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[Crossref] [PubMed]

S. Takeda, M. Fuwa, P. van Loock, and A. Furusawa, “Entanglement swapping between discrete and continuous variables,” Phys. Rev. Lett. 114, 100501 (2015).
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R. E. S. Polkinghorne and T. C. Ralph, “Continuous variable entanglement swapping,” Phys. Rev. Lett. 83, 2095-2099 (1999).
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H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910-913 (2001).
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D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantum computation,” Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287-4290 (1993).
[Crossref]

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).
[Crossref]

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118, 230501 (2017).
[Crossref] [PubMed]

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482-3485 (2000).
[Crossref] [PubMed]

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. I. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106, 240504 (2011).
[Crossref] [PubMed]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
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N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,”, Phys. Rev. Lett. 97110501 (2006).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Graph representation of principle for entanglement swapping between two four-mode Gaussian cluster states. Nodes and edges in the graph denote optical modes and the nearest neighbor interaction between modes. (a) The representation of entanglement swapping between two four-mode square Gaussian cluster states for scheme I. (b) The representation of entanglement swapping between a linear and a star-shape four-mode Gaussian cluster state for scheme II.
Fig. 2
Fig. 2 Schematic of two entanglement swapping schemes. (a) Scheme for entanglement swapping between two four-mode square Gaussian cluster states. The joint measurement is performed on optical modes A ^ 1 and B ^ 1 coming from two cluster states A and B, respectively. The measurement results are fedforward to the remaining quantum modes of multipartite entangled states A and B through classical channels, respectively. (b) Scheme for entanglement swapping between a four-mode linear and a star-shape Gaussian cluster states. The joint measurement is performed on optical modes D ^ 1 and E ^ 2 coming from two multipartite entangled states D and E, respectively. The measurement results are fedforward to the remaining quantum modes of multipartite entangled state E through classical channels. EOMX and EOMP, amplitude and phase electro-optical modulators; HD, homodyne detector; LO, local beam. The power splitter is used to split the output photocurrent from the homodyne detector.
Fig. 3
Fig. 3 The inseparability criteria for the obtained six-mode cluster entangled state in entanglement swapping scheme I. (a) - (h) are corresponding to (a) - (h) inequalities in Eq. (6), respectively. Green dash-dotted lines and blue dashed lines represent the results with unit and optimal gain G1 in classical channels, respectively, when the unit gains g C i in the inseparability criteria are chosen. Red solid lines represent the results with both optimal gains G1 and g C i.
Fig. 4
Fig. 4 The inseparability criteria for the obtained six-mode cluster entangled state in swapping scheme II. (a) - (e) are corresponding to (a) - (e) inequalities in Eq. (14), respectively. Green dash-dotted lines and blue dashed lines represent the results with unit and optimal gain G2 in classical channels, respectively, when the unit gains g F i in the inseparability criteria are chosen. Red solid lines represent the results with both optimal gains G2 and g F i.

Equations (27)

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

( p ^ a b N a x ^ b ) 0 , a G
U A = U B = [ 1 2 2 5 i 1 10 0 1 2 2 5 i 1 10 0 0 i 1 10 2 5 1 2 0 i 1 10 2 5 1 2 ] .
p ^ A 1 x ^ A 3 x ^ A 4 = 1 2 p ^ a 1 ( 0 ) e r 5 2 x ^ a 3 ( 0 ) e r , p ^ A 2 x ^ A 3 x ^ A 4 = 1 2 p ^ a 1 ( 0 ) e r 5 2 x ^ a 3 ( 0 ) e r , p ^ A 3 x ^ A 1 x ^ A 2 = 1 2 p ^ a 4 ( 0 ) e r + 5 2 x ^ a 2 ( 0 ) e r , p ^ A 4 x ^ A 1 x ^ A 2 = 1 2 p ^ a 4 ( 0 ) e r 5 2 x ^ a 2 ( 0 ) e r ,
x ^ μ 1 = 1 2 ( x ^ A 1 p ^ B 1 ) , p ^ ν 1 = 1 2 ( p ^ A 1 x ^ B 1 ) ,
x ^ C 1 = x ^ A 3 , p ^ C 1 = p ^ A 3 2 G 1 x ^ μ 1 , x ^ C 2 = x ^ A 2 , p ^ C 2 = p ^ A 2 , x ^ C 3 = x ^ A 4 , p ^ C 3 = p ^ A 4 2 G 1 x ^ μ 1 , x ^ C 4 = x ^ B 4 , p ^ C 4 = p ^ B 4 + 2 G 1 p ^ ν 1 , x ^ C 5 = x ^ B 2 , p ^ C 5 = p ^ B 2 , x ^ C 6 = x ^ B 3 , p ^ C 6 = p ^ B 3 + 2 G 1 p ^ ν 1 ,
Δ 2 ( p ^ C 1 x ^ C 2 g C 1 x ^ C 6 g C 1 x ^ C 4 ) + Δ 2 ( p ^ C 2 x ^ C 1 g C 1 x ^ C 3 ) < 1 ,
Δ 2 ( p ^ C 1 g C 2 x ^ C 2 g C 2 x ^ C 6 x ^ C 4 ) + Δ 2 ( p ^ C 4 g C 2 x ^ C 3 g C 2 x ^ C 5 x ^ C 1 ) < 1 ,
Δ 2 ( p ^ C 1 g C 3 x ^ C 2 x ^ C 6 g C 3 x ^ C 4 ) + Δ 2 ( p ^ C 6 x ^ C 1 g C 3 x ^ C 5 g C 3 x ^ C 3 ) < 1 ,
Δ 2 ( p ^ C 2 g C 4 x ^ C 1 x ^ C 3 ) + Δ 2 ( p ^ C 3 x ^ C 2 g C 4 x ^ C 4 g C 4 x ^ C 6 ) < 1 ,
Δ 2 ( p ^ C 3 g C 5 x ^ C 2 x ^ C 4 g C 5 x ^ C 6 ) + Δ 2 ( p ^ C 4 x ^ C 3 g C 5 x ^ C 5 g C 5 x ^ C 1 ) < 1 ,
Δ 2 ( p ^ C 3 g C 6 x ^ C 2 g C 6 x ^ C 4 x ^ C 6 ) + Δ 2 ( p ^ C 6 g C 6 x ^ C 1 g C 6 x ^ C 5 x ^ C 3 ) < 1 ,
Δ 2 ( p ^ C 4 g C 7 x ^ C 3 x ^ C 5 g C 7 x ^ C 1 ) + Δ 2 ( p ^ C 5 x ^ C 4 g C 7 x ^ C 6 ) < 1 ,
Δ 2 ( p ^ C 5 g C 8 x ^ C 4 x ^ C 6 ) + Δ 2 ( p ^ C 6 g C 8 x ^ C 1 x ^ C 5 g C 8 x ^ C 3 ) < 1 .
G 1 = 1 22 e 4 r + 21 e 8 r 11 + 68 e 4 r + 21 e 8 r .
g C 1 = g C 4 = g C 7 = g C 8 = 21 e 8 r 12 e 4 r 9 46 + 83 e 4 r + 21 e 8 r , g C 2 = g C 3 = g C 5 = g C 6 = 21 e 8 r 12 e 4 r 9 11 + 68 e 4 r + 21 e 8 r ,
U D = [ 1 2 2 5 i 1 10 0 i 1 2 i 2 5 1 10 0 0 1 10 i 2 5 i 1 2 0 i 1 10 2 5 1 2 ] ,
U E = [ i 1 2 i 2 i 2 0 1 2 1 2 1 2 0 0 1 2 1 2 i 1 2 0 1 2 1 2 i 1 2 ] ,
p ^ D 1 x ^ D 2 = 2 p ^ d 1 ( 0 ) e r , p ^ D 2 x ^ D 1 x ^ D 3 = 1 2 p ^ d 4 ( 0 ) e r 5 2 x ^ d 2 ( 0 ) e r , p ^ D 3 x ^ D 2 x ^ D 4 = 1 2 p ^ d 1 ( 0 ) e r 5 2 x ^ d 3 ( 0 ) e r , p ^ D 4 x ^ D 3 = 2 p ^ d 4 ( 0 ) e r , p ^ E 1 x ^ E 2 x ^ E 3 x ^ E 4 = 2 x ^ e 2 ( 0 ) e r , p ^ E 2 x ^ E 1 = 2 p ^ e 1 ( 0 ) e r , p ^ E 3 x ^ E 1 = 1 2 p ^ e 1 ( 0 ) e r + p ^ e 3 ( 0 ) e r + 1 2 x ^ e 4 ( 0 ) e 4 , p ^ E 4 x ^ E 1 = 1 2 p ^ e 1 ( 0 ) e r + p ^ e 3 ( 0 ) e r 1 2 x ^ e 4 ( 0 ) e r ,
x ^ μ 2 = 1 2 ( x ^ D 1 p ^ E 2 ) , p ^ ν 2 = 1 2 ( p ^ D 1 x ^ E 2 ) ,
x ^ F 1 = x ^ D 4 , p ^ F 1 = p ^ D 4 , x ^ F 2 = x ^ D 3 , p ^ F 2 = p ^ D 3 , x ^ F 3 = x ^ D 2 , p ^ F 3 = p ^ D 2 , x ^ F 4 = x ^ E 1 + 2 G 2 x ^ μ 2 , p ^ F 4 = p ^ E 1 + 2 G 2 p ^ ν 2 , x ^ F 5 = x ^ E 3 , p ^ F 5 = p ^ E 3 + 2 G 2 x ^ μ 2 , x ^ F 6 = x ^ E 4 , p ^ F 6 = p ^ E 4 + 2 G 2 x ^ μ 2 ,
Δ 2 ( p ^ F 1 x ^ F 2 ) + Δ 2 ( p ^ F 2 x ^ F 1 g F 1 x ^ F 3 ) < 1 ,
Δ 2 ( p ^ F 2 g F 2 x ^ F 1 x ^ F 3 ) + Δ 2 ( p ^ F 3 x ^ F 2 g F 2 x ^ F 4 ) < 1 ,
Δ 2 ( p ^ F 3 g F 3 x ^ F 2 x ^ F 4 ) + Δ 2 ( p ^ F 4 x ^ F 3 g F 3 x ^ F 5 g F 3 x ^ F 6 ) < 1 ,
Δ 2 ( p ^ F 4 g F 4 x ^ F 3 x ^ F 5 g F 4 x ^ F 6 ) + Δ 2 ( p ^ F 5 x ^ F 4 ) < 1 ,
Δ 2 ( p ^ F 4 g F 5 x ^ F 3 g F 5 x ^ F 5 x ^ F 6 ) + Δ 2 ( p ^ F 6 x ^ F 4 ) < 1 .
G 2 = 17 e 4 r 17 17 e 4 r + 23 .
g F 1 = 2 e 4 r 2 3 + 2 e 4 r , g F 2 = 3 ( e 4 r 1 ) ( 3 + 17 e 4 r ) 81 + 68 e 4 r + 51 e 8 r , g F 3 = 7 ( e 4 r 1 ) ( 3 + 17 e 4 r ) ( 8 + 7 e 4 r ) + ( 23 + 17 e 4 r ) , g F 4 = g F 5 = ( e 4 r 1 ) ( 9 + 39 e 4 r ) ( 23 + 17 e 4 r ) + ( 17 + 23 e 4 r ) ,

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