Abstract

We propose an effective protocol for preparation of four-photon polarization-entangled decoherence-free states with quantum nondemolition detectors. The protocol is based on optical elements, single polarization photons, and cross-Kerr nonlinearity, which are feasible with existing experimental technology. Compared with previous protocols, the present one is to replace the entangled-state resources with much simpler single-photon resources and has a higher success probability. All these advantages make this protocol more efficient and more convenient than others in the applications in quantum communication.

© 2013 Optical Society of America

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    [CrossRef]
  29. Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. 90, 207901 (2003).
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    [CrossRef]
  31. 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]
  32. 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]
  33. J. W. Pan, C. Simon, C. Brukner, and A. Zellinger, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001).
    [CrossRef]
  34. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002).
    [CrossRef]
  35. 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]
  36. X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
    [CrossRef]
  37. 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]
  38. F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
    [CrossRef]
  39. 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]
  40. C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).
  41. F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
    [CrossRef]
  42. C. Wang, Y. B. Sheng, X. H. Li, F. G. Deng, W. Zhang, and G. L. Long, “Efficient entanglement purification for doubly entangled photon state,” Sci. China Ser. E 52, 3464–3467 (2009).
    [CrossRef]
  43. L. M. Duan and G. C. Guo, “Preserving coherence in quantum computation by pairing quantum bits,” Phys. Rev. Lett. 79, 1953–1956 (1997).
    [CrossRef]
  44. J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, “Theory of decoherence-free fault-tolerant universal quantum computation,” Phys. Rev. A 63, 042307 (2001).
    [CrossRef]
  45. J. B. Altepeter, P. G. Hadley, S. M. Wendelken, A. J. Berglund, and P. G. Kwiat, “Experimental investigation of a two-qubit decoherence-free subspace,” Phys. Rev. Lett. 92, 147901 (2004).
    [CrossRef]
  46. M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer, A. Cabello, and H. Weinfurter, “Decoherence-free quantum information processing with four-photon entangled states,” Phys. Rev. Lett. 92, 107901 (2004).
    [CrossRef]
  47. X. B. Zou, J. Shu, and G. C. Guo, “Simple scheme for generating four-photon polarization-entangled decoherence-free states using spontaneous parametric down-conversions,” Phys. Rev. A 73, 054301 (2006).
    [CrossRef]
  48. Y. X. Gong, X. B. Zou, X. L. Niu, J. Li, Y. F. Huang, and G. C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
    [CrossRef]
  49. Y. Xia, J. Song, H. S. Song, and S. Zhang, “Controlled generation of four-photon polarization-entangled decoherence-free states with conventional photon detectors,” J. Opt. Soc. Am. B 26, 129–132 (2009).
    [CrossRef]
  50. Y. Xia, J. Song, Z. B. Yang, and S. B. zheng, “Generation of four-photon polarization-entangled decoherence-free states within a network,” Appl. Phys. B 99, 651–656 (2010).
    [CrossRef]
  51. Y. Xia, J. Song, P. M. Lu, and H. S. Song, “Generation of four-atom entangled decoherence-free states by interference of polarized photons,” J. Mod. Opt. 56, 1545–1549 (2009).
    [CrossRef]
  52. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
    [CrossRef]
  53. Y. Xia, J. Song, P. M. Lu, and H. S. Song, “Efficient implementation of the two-qubit controlled phase gate with cross-Kerr nonlinearity,” J. Phys. B 44, 025503 (2011).
    [CrossRef]
  54. Y. Xia, Q. Q. Chen, J. Song, and H. S. Song, “Efficient hyperentangled Greenberger–Horne–Zeilinger states analysis with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1029–1037 (2012).
    [CrossRef]
  55. Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010).
    [CrossRef]
  56. G. L. Long, “General quantum interference principle and duality computer,” Commun. Theor. Phys. 45, 825–844 (2006).
    [CrossRef]
  57. X. B. Zou, K. Pahike, and W. Mathis, “Generation of an entangled four-photon W state,” Phys. Rev. A 66, 044302 (2002).
    [CrossRef]
  58. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [CrossRef]
  59. J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
    [CrossRef]
  60. J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 1–17 (2007).
    [CrossRef]
  61. S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
    [CrossRef]
  62. H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, “Optimized phase switching using a single-atom nonlinearity,” J. Opt. B Quantum Semiclass. Opt 5, 218–221 (2003).
    [CrossRef]
  63. P. Kok, “Effects of self-phase-modulation on weak nonlinear optical quantum gates,” Phys. Rev. A 77, 013808 (2008).
    [CrossRef]
  64. Q. Lin and B. He, “Single-photon logic gates using minimal resources,” Phys. Rev. A 80, 042310 (2009).
    [CrossRef]
  65. Q. Lin, B. He, J. A. Bergou, and Y. H. Ren, “Processing multiphoton states through operation on a single photon: methods and applications,” Phys. Rev. A 80, 042311 (2009).
    [CrossRef]
  66. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
    [CrossRef]
  67. Y. Xia, J. Song, and H. S. Song, “Linear optical protocol for preparation of N-photon Greenberger–Horne–Zeilinger state with conventional photon detectors,” Appl. Phys. Lett. 92, 021127 (2008).
    [CrossRef]
  68. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter, “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. 92, 077901 (2004).
    [CrossRef]
  69. J. W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement,” Nature 403, 515–519 (2000).
    [CrossRef]
  70. D. Bouwmeester, J. W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger–Horne–Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
    [CrossRef]

2012

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]

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

Y. Xia, Q. Q. Chen, J. Song, and H. S. Song, “Efficient hyperentangled Greenberger–Horne–Zeilinger states analysis with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1029–1037 (2012).
[CrossRef]

2011

Y. Xia, J. Song, P. M. Lu, and H. S. Song, “Efficient implementation of the two-qubit controlled phase gate with cross-Kerr nonlinearity,” J. Phys. B 44, 025503 (2011).
[CrossRef]

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

F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
[CrossRef]

2010

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (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 and F. G. Deng, “Efficient quantum entanglement distribution over an arbitrary collective-noise channel,” Phys. Rev. A 81, 042332 (2010).
[CrossRef]

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

Y. Xia, J. Song, Z. B. Yang, and S. B. zheng, “Generation of four-photon polarization-entangled decoherence-free states within a network,” Appl. Phys. B 99, 651–656 (2010).
[CrossRef]

2009

Y. Xia, J. Song, P. M. Lu, and H. S. Song, “Generation of four-atom entangled decoherence-free states by interference of polarized photons,” J. Mod. Opt. 56, 1545–1549 (2009).
[CrossRef]

Y. Xia, J. Song, H. S. Song, and S. Zhang, “Controlled generation of four-photon polarization-entangled decoherence-free states with conventional photon detectors,” J. Opt. Soc. Am. B 26, 129–132 (2009).
[CrossRef]

C. Wang, L. Xiao, W. Y. Wang, G. Y. Zhang, and G. L. Long, “Quantum key distribution using polarization and frequency hyperentangled photons,” J. Opt. Soc. Am. B 26, 2072–2076 (2009).
[CrossRef]

Q. Lin and B. He, “Single-photon logic gates using minimal resources,” Phys. Rev. A 80, 042310 (2009).
[CrossRef]

Q. Lin, B. He, J. A. Bergou, and Y. H. Ren, “Processing multiphoton states through operation on a single photon: methods and applications,” Phys. Rev. A 80, 042311 (2009).
[CrossRef]

C. Wang, Y. B. Sheng, X. H. Li, F. G. Deng, W. Zhang, and G. L. Long, “Efficient entanglement purification for doubly entangled photon state,” Sci. China Ser. E 52, 3464–3467 (2009).
[CrossRef]

2008

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. Xia, J. Song, and H. S. Song, “Linear optical protocol for preparation of N-photon Greenberger–Horne–Zeilinger state with conventional photon detectors,” Appl. Phys. Lett. 92, 021127 (2008).
[CrossRef]

P. Kok, “Effects of self-phase-modulation on weak nonlinear optical quantum gates,” Phys. Rev. A 77, 013808 (2008).
[CrossRef]

Y. X. Gong, X. B. Zou, X. L. Niu, J. Li, Y. F. Huang, and G. C. Guo, “Generation of arbitrary four-photon polarization-entangled decoherence-free states,” Phys. Rev. A 77, 042317 (2008).
[CrossRef]

2007

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

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 1–17 (2007).
[CrossRef]

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

2006

J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
[CrossRef]

G. L. Long, “General quantum interference principle and duality computer,” Commun. Theor. Phys. 45, 825–844 (2006).
[CrossRef]

X. B. Zou, J. Shu, and G. C. Guo, “Simple scheme for generating four-photon polarization-entangled decoherence-free states using spontaneous parametric down-conversions,” Phys. Rev. A 73, 054301 (2006).
[CrossRef]

2005

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

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

D. Kalamidas, “Single-photon quantum error rejection and correction with linear optics,” Phys. Lett. A 343, 331–335 (2005).
[CrossRef]

2004

H. Kim, Y. M. Cheong, and H. W. Lee, “Generalized measurement and conclusive teleportation with nonmaximal entanglement,” Phys. Rev. A 70, 012309 (2004).
[CrossRef]

J. B. Altepeter, P. G. Hadley, S. M. Wendelken, A. J. Berglund, and P. G. Kwiat, “Experimental investigation of a two-qubit decoherence-free subspace,” Phys. Rev. Lett. 92, 147901 (2004).
[CrossRef]

M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer, A. Cabello, and H. Weinfurter, “Decoherence-free quantum information processing with four-photon entangled states,” Phys. Rev. Lett. 92, 107901 (2004).
[CrossRef]

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter, “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. 92, 077901 (2004).
[CrossRef]

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

2003

H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, “Optimized phase switching using a single-atom nonlinearity,” J. Opt. B Quantum Semiclass. Opt 5, 218–221 (2003).
[CrossRef]

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

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

F. G. Deng and G. L. Long, “Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block,” Phys. Rev. A 68, 042317 (2003).
[CrossRef]

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

2002

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

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

X. B. Zou, K. Pahike, and W. Mathis, “Generation of an entangled four-photon W state,” Phys. Rev. A 66, 044302 (2002).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

2001

J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, “Theory of decoherence-free fault-tolerant universal quantum computation,” Phys. Rev. A 63, 042307 (2001).
[CrossRef]

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

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

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

2000

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

J. W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef]

1999

D. Bouwmeester, J. W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger–Horne–Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
[CrossRef]

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

L. Viola, E. Knill, and S. Lloyd, “Dynamical decoupling of open quantum systems,” Phys. Rev. Lett. 82, 2417–2421 (1999).
[CrossRef]

1997

L. M. Duan and G. C. Guo, “Preserving coherence in quantum computation by pairing quantum bits,” Phys. Rev. Lett. 79, 1953–1956 (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, 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]

A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett. 77, 793–797 (1996).
[CrossRef]

R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum error correcting code,” Phys. Rev. Lett. 77, 198–201(1996).
[CrossRef]

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[CrossRef]

1995

P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A 52, R2493 (1995).
[CrossRef]

1993

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram for preparation of four-photon polarization-entangled decoherence-free states with cross-Kerr nonlinearity.

Fig. 2.
Fig. 2.

Setup of the QND x , x { 1 , 2 , 3 , 4 } .

Fig. 3.
Fig. 3.

Experimental setup to change the bunching photons into the unbunching state.

Fig. 4.
Fig. 4.

Setup of the QND y , y { 1 , 2 } .

Fig. 5.
Fig. 5.

Experimental setup to change the two bunching photons in modes c 1 or c 2 to two different modes k 1 and k 2 .

Fig. 6.
Fig. 6.

Experimental setup to change the two bunching photons in modes k 1 or k 2 to two different modes f 1 and f 2 .

Fig. 7.
Fig. 7.

Overall probability of success P total versus m and n . P total increases when increasing iteration numbers n and m . Here we set 0 n ( m ) 10 , cos θ = sin θ = 1 / 2 .

Equations (28)

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| Φ 0 = α | Ψ 0 + β | Ψ 1
| Ψ 0 = 1 2 ( | H 1 | V 2 | V 1 | H 2 ) ( | H 3 | V 4 | V 3 | H 4 ) ,
| Ψ 1 = 1 2 3 ( 2 | V 1 | V 2 | H 3 | H 4 + 2 | H 1 | H 2 | V 3 | V 4 | V 1 | H 2 | V 3 | H 4 | V 1 | H 2 | H 3 | V 4 | H 1 | V 2 | H 3 | V 4 | H 1 | V 2 | V 3 | H 4 ) ,
| ϕ 1 | ϕ 2 | ϕ 3 | ϕ 4 = 1 4 ( | H + | V ) 1 ( | H + | V ) 2 ( | H + | V ) 3 ( | H + | V ) 4 ,
| ϖ = | ϕ 1 | ϕ 2 | ϕ 3 | ϕ 4 PBS 1 , PBS 2 1 2 ( | H b 1 | H b 2 + | H b 2 | V b 2 + | V b 1 | H b 1 + | V b 1 | V b 2 ) 1 2 ( | H b 3 | H b 4 + | H b 3 | V b 3 + | V b 4 | H b 4 + | V b 3 | V b 4 ) .
U c k | φ | α = e i H QND t / 1 2 ( | 0 b x + | 1 b x ) | α x = 1 2 ( | 0 b x | α 1 + | 1 b x | α e i θ x ) ,
| ϖ | α x = 1 2 ( | H b 1 | H b 2 + | H b 2 | V b 2 + | V b 1 | H b 1 + | V b 1 | V b 2 ) 1 2 ( | H b 3 | H b 4 + | H b 3 | V b 3 + | V b 4 | H b 4 + | V b 3 | V b 4 ) | α x QND x 1 2 [ ( | H b 1 | H b 2 + | V b 1 | V b 2 ) | α e i ( θ 1 + θ 2 ) x + | H b 2 | V b 2 | α e 2 i θ 2 x + | V b 1 | H b 1 | α e 2 i θ 1 x ] 1 2 [ ( | H b 3 | H b 4 + | V b 3 | V b 4 ) | α e i ( θ 3 + θ 4 ) x + | H b 3 | V b 3 | α e 2 i θ 3 x + | V b 4 | H b 4 | α e 2 i θ 4 x ] .
| V b 1 | H b 1 PBS 3 | V e 2 | H e 1 QWP 1 , QWP 2 1 2 ( | H e 2 | V e 2 ) ( | H e 1 + | V e 1 ) PBS 4 1 2 ( | H b 5 | V b 1 ) ( | H b 1 + | V b 5 ) 1 2 ( | H b 5 | H b 1 + | H b 5 | V b 5 | V b 1 | H b 1 | V b 1 | V b 5 ) QND 1 , QND 2 1 2 [ ( | H b 1 | H b 5 | V b 1 | V b 5 ) | α e i ( θ 1 + θ 2 ) x + | H b 5 | V b 5 | α e 2 i θ 2 x | V b 1 | H b 1 | α e 2 i θ 3 x ] π / 2 phaseshifter P 1 2 [ ( | H b 1 | H b 5 + | V b 1 | V b 5 ) | α e i ( θ 1 + θ 2 ) x + | H b 5 | V b 5 | α e 2 i θ 2 x + | V b 1 | H b 1 | α e 2 i θ 3 x ] .
P 1 = 1 2 + 1 2 [ 1 2 + 1 2 [ 1 2 + 1 2 [ 1 2 + ] ] ] = 1 ( 1 2 ) n + 1 ,
| ϖ 1 = 1 2 ( | H b 1 | V b 2 | V b 1 | H b 2 ) 1 2 ( | H b 3 | V b 4 | V b 3 | H b 4 ) ,
P 2 = [ 1 ( 1 2 ) n + 1 ] 2 ,
a b 2 , k + ( cos θ ) a c 1 , k + + ( sin θ ) a c 2 , k + ,
a b 3 , k + ( cos θ ) a c 2 , k + ( sin θ ) a c 1 , k + ,
| ϖ 2 = 1 2 ( | H b 1 | V b 2 | V b 1 | H b 2 ) ( | H b 3 | V b 4 | V b 3 | H b 4 ) BS 1 2 [ cos 2 θ | H b 1 V c 1 H c 2 V b 4 cos θ sin θ | H b 1 V c 1 H c 1 V b 4 cos 2 θ | H b 1 V c 1 V c 2 H b 4 + cos θ sin θ | H b 1 V c 1 V c 1 H b 4 + cos θ sin θ | H b 1 V c 2 H c 2 V b 4 sin 2 θ | H b 1 V c 2 H c 1 V b 4 sin θ cos θ | H b 1 V c 2 V c 2 H b 4 + sin 2 θ | H b 1 V c 2 V c 1 H b 4 cos 2 θ | V b 1 H c 1 H c 2 V b 4 + cos θ sin θ | V b 1 H c 1 H c 1 V b 4 + cos 2 θ | V b 1 H c 1 V c 2 H b 4 cos θ sin θ | V b 1 H c 1 V c 1 H b 4 cos θ sin θ | V b 1 H c 2 H c 2 V b 4 + sin 2 θ | V b 1 H c 2 H c 1 V b 4 + sin θ cos θ | V b 1 H c 2 V c 2 H b 4 sin 2 θ | V b 1 H c 2 V c 1 H b 4 ] .
| ϖ 3 QND 5 , QND 6 1 2 [ cos 2 θ | H b 1 V c 1 H c 2 V b 4 | α e i ( θ 5 + θ 6 ) y cos θ sin θ | H b 1 V c 1 H c 1 V b 4 | α e 2 i θ 5 y cos 2 θ | H b 1 V c 1 V c 2 H b 4 | α e i ( θ 5 + θ 6 ) y + cos θ sin θ | H b 1 V c 1 V c 1 H b 4 | α e 2 i θ 5 y + cos θ sin θ | H b 1 V c 2 H c 2 V b 4 | α e 2 i θ 6 y sin 2 θ | H b 1 V c 2 H c 1 V b 4 | α e i ( θ 5 + θ 6 ) y sin θ cos θ | H b 1 V c 2 V c 2 H b 4 | α e 2 i θ 6 y + sin 2 θ | H b 1 V c 2 V c 1 H b 4 | α e i ( θ 5 + θ 6 ) y cos 2 θ | V b 1 H c 1 H c 2 V b 4 | α e i ( θ 5 + θ 6 ) y + cos θ sin θ | V b 1 H c 1 H c 1 V b 4 | α e 2 i θ 5 y + cos 2 θ | V b 1 H c 1 V c 2 H b 4 | α e i ( θ 5 + θ 6 ) y cos θ sin θ | V b 1 H c 1 V c 1 H b 4 | α e 2 i θ 5 y cos θ sin θ | V b 1 H c 2 H c 2 V b 4 | α e 2 i θ 6 y + sin 2 θ | V b 1 H c 2 H c 1 V b 4 | α e i ( θ 5 + θ 6 ) y + sin θ cos θ | V b 1 H c 2 V c 2 H b 4 | α e 2 i θ 6 y sin 2 θ | V b 1 H c 2 V c 1 H b 4 | α e i ( θ 5 + θ 6 ) y ] .
| Φ = cos 2 θ | H b 1 V c 1 H c 2 V b 4 cos 2 θ | H b 1 V c 1 V c 2 H b 4 sin 2 θ | H b 1 V c 2 H c 1 V b 4 + sin 2 θ | H b 1 V c 2 V c 1 H b 4 cos 2 θ | V b 1 H c 1 H c 2 V b 4 + cos 2 θ | V b 1 H c 1 V c 2 H b 4 + sin 2 θ | V b 1 H c 2 H c 1 V b 4 sin 2 θ | V b 1 H c 2 V c 1 H b 4 ,
| ζ = cos 2 θ ( | H b 1 V c 1 | V b 1 H c 1 ) ( | H c 2 V b 4 | V c 2 H b 4 ) sin 2 θ ( | H b 1 V c 2 | H b 1 V c 2 ) ( | H c 1 V b 4 | V c 1 H b 4 ) = ( cos 2 θ 1 2 sin 2 θ ) ( | H b 1 V c 1 | V b 1 H c 1 ) ( | H c 2 V b 4 | V c 2 H b 4 ) sin 2 θ [ | V b 1 V c 1 H c 2 H b 4 + | H b 1 H c 1 V c 2 V b 4 1 2 ( | H b 1 V c 2 + | H b 1 V c 2 ) ( | H c 1 V b 4 + | V c 1 H b 4 ) ] = 1 4 3 sin 2 2 θ [ ( 2 3 sin 2 θ ) | Ψ 0 3 sin 2 θ | Ψ 1 ] ,
α = 2 3 sin 2 θ 4 3 sin 2 2 θ ,
β = 3 sin 2 θ 4 3 sin 2 2 θ .
| η = cos θ sin θ | H b 1 V c 1 H c 1 V b 4 + cos θ sin θ | H b 1 V c 1 V c 1 H b 4 + cos θ sin θ | H b 1 V c 2 H c 2 V b 4 cos θ sin θ | H b 1 V c 2 V c 2 H b 4 + cos θ sin θ | V b 1 H c 1 H c 1 V b 4 cos θ sin θ | V b 1 H c 1 V c 1 H b 4 cos θ sin θ | V b 1 H c 2 H c 2 V b 4 + cos θ sin θ | V b 1 H c 2 V c 2 H b 4 = cos θ sin θ [ | H b 1 V c 1 H c 1 V b 4 | H b 1 V c 1 V c 1 H b 4 | H b 1 V c 2 H c 2 V b 4 + | H b 1 V c 2 V c 2 H b 4 | V b 1 H c 1 H c 1 V b 4 + | V b 1 H c 1 V c 1 H b 4 + | V b 1 H c 2 H c 2 V b 4 | V b 1 H c 2 V c 2 H b 4 ] .
| η 1 = cos θ sin θ ( | H b 1 V k 2 H k 1 V b 4 | α e i ( θ 1 + θ 2 ) x | H b 1 V k 2 V k 2 H b 4 | α e 2 i θ 2 x | H b 1 V k 2 H k 1 V b 4 | α e i ( θ 1 + θ 2 ) x + | H b 1 V k 2 V k 2 H b 4 | α e 2 i θ 2 x | V b 1 H k 1 H k 1 V b 4 | α e 2 i θ 1 x + | V b 1 H k 1 V k 2 H b 4 | α e i ( θ 1 + θ 2 ) x + | V b 1 H k 1 H k 1 V b 4 | α e 2 i θ 1 x | V b 1 H k 1 V k 2 H b 4 | α e i ( θ 1 + θ 2 ) x ) .
P 4 = cos 2 θ sin 2 θ .
| V k 2 V k 2 | α x BS 50 50 1 2 ( | V f 1 + | V f 2 ) ( | V f 1 + | V f 2 ) | α x QND 1 , QND 2 1 2 ( | V f 1 V f 1 | α e 2 i θ 1 x + | V f 2 V f 1 | α e i ( θ 1 + θ 2 ) x + | V f 2 V f 2 | α e 2 i θ 2 x + | V f 1 V f 2 | α e i ( θ 1 + θ 2 ) x ) .
P 5 = 1 ( 1 2 ) m ,
P 6 = cos 2 θ sin 2 θ [ 1 ( 1 2 ) m ] ,
| η 2 = cos θ sin θ ( | H b 1 V c 1 H c 2 V b 4 | H b 1 V c 1 V c 2 H b 4 | H b 1 V c 2 H c 1 V b 4 + | H b 1 V c 2 V c 1 H b 4 | V b 1 H c 1 H c 2 V b 4 + | V b 1 H c 1 V c 2 H b 4 + | V b 1 H c 2 H c 1 V b 4 | V b 1 H c 2 V c 1 H b 4 ) = cos θ sin θ [ ( | H b 1 V c 1 | V b 1 H c 1 ) ( | H c 2 V b 4 | V c 2 H b 4 ) ( | H b 1 V c 2 | H b 1 V c 2 ) ( | H c 1 V b 4 | V c 1 H b 4 ) ] = cos θ sin θ { 1 2 ( | H b 1 V c 1 | V b 1 H c 1 ) ( | H c 2 V b 4 | V c 2 H b 4 ) [ | V b 1 V c 1 H c 2 H b 4 + | H b 1 H c 1 V c 2 V b 4 1 2 ( | H b 1 V c 2 + | H b 1 V c 2 ) ( | H c 1 V b 4 + | V c 1 H b 4 ) ] } = 1 2 ( | Ψ 0 + 3 | Ψ 1 ) ,
P total = P 2 ( P 3 + P 7 ) = [ 1 ( 1 2 ) n + 1 ] 2 × { ( cos 4 θ + sin 4 θ ) + 2 cos 2 θ sin 2 θ [ 1 ( 1 2 ) m ] } .
P total = [ 1 ( 1 2 ) n + 1 ] 2 [ 1 ( 1 2 ) m + 1 ] ,

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