Abstract

We propose methods for generating Bell-type entangled coherent states for quantum information processing. The protocols are completed using linear optical elements, single-photon detectors, and cross-Kerr nonlinearity. We investigate the decoherence effect on our protocols; photon loss is found to affect only the probability, not the fidelity, of the generation process. The entanglement of the generated entangled states is also analyzed using concurrence. Finally, the feasibility of our scheme is discussed using the continuous-time cross-phase modulation model. In the weak nonlinearity region, on the condition that θ2|αp|21, θ1, and σ21, our scheme is applicable theoretically.

© 2013 Optical Society of America

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  9. N. B. An, “Optimal processing of quantum information via W-type entangled coherent states,” Phys. Rev. A 69, 022315 (2004).
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  32. H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
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  33. H. Jeong, M. S. Kim, T. C. Ralph, and B. S. Ham, “Generation of macroscopic superposition states with small nonlinearity,” Phys. Rev. A 70, 061801(R) (2004).
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  34. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical Controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
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    [CrossRef]
  36. A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent states with inefficient photon detection,” Phys. Rev. A 70, 020101 (2004).
    [CrossRef]
  37. B. Wang and L. M. Duan, “Engineering superpositions of coherent states in coherent optical pulses through cavity-assisted interaction,” Phys. Rev. A 72, 022320 (2005).
    [CrossRef]
  38. B. He, M. Nadeem, and J. A. Bergou, “Scheme for generating coherent-state superpositions with realistic cross-Kerr nonlinearity,” Phys. Rev. A 79, 035802 (2009).
    [CrossRef]
  39. S. J. van Enk, “Entanglement capabilities in infinite dimensions: multidimensional entangled coherent states,” Phys. Rev. Lett. 91, 017902 (2003).
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  40. B. He, Y. Ren, and J. A. Bergou, “Creation of high-quality long-distance entanglement with flexible resources,” Phys. Rev. A 79, 052323 (2009).
    [CrossRef]
  41. B. He, Y. Ren, and J. A. Bergou, “Universal entangler with photon pairs in arbitrary states,” J. Phys. B 43, 025502 (2010).
    [CrossRef]
  42. C. C. Gerry, “Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium,” Phys. Rev. A 59, 4095–4098 (1999).
    [CrossRef]
  43. Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79, 022301 (2009).
    [CrossRef]
  44. Q. Lin and B. He, “Single-photon logic gates using minimal resources,” Phys. Rev. A 80, 042310 (2009).
    [CrossRef]
  45. F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
    [CrossRef]
  46. C. Wang, Y. S. Li, and L. Hao, “Optical implementation of quantum random walks using weak cross-Kerr media,” Chinese Sci. Bull. 56, 2088–2091 (2011).
    [CrossRef]
  47. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2012).
    [CrossRef]
  48. S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
    [CrossRef]
  49. R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
    [CrossRef]
  50. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
    [CrossRef]
  51. H. Weinfurter and M. Zukowski, “Four-photon entanglement from down-conversion,” Phys. Rev. A 64, 010102(R) (2001).
    [CrossRef]
  52. X.-H. Bao, Y. Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
    [CrossRef]
  53. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
    [CrossRef]
  54. J. H. Shapiro, “Single-photon Kerr nonlinearities do not help quantum computation,” Phys. Rev. A 73, 062305 (2006).
    [CrossRef]
  55. J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
    [CrossRef]
  56. J. Gea-Banacloche, “Impossibility of large phase shifts via the giant Kerr effect with single-photon wave packets,” Phys. Rev. A 81, 043823 (2010).
    [CrossRef]
  57. I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
    [CrossRef]
  58. M. Siomau, A. A. Kamli, S. A. Moiseev, and B. C. Sanders, “Entanglement creation with negative index metamaterials,” Phys. Rev. A 85, 050303(R) (2012).
    [CrossRef]
  59. S. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
    [CrossRef]
  60. D. Braje, V. Balić, G. Yin, and S. Harris, “Low-light-level non-linear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
    [CrossRef]
  61. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995).
    [CrossRef]
  62. P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum nondemolition measurements in optics,” Nature 396, 537–542 (1998).
    [CrossRef]

2013 (1)

Y. B. Sheng and L. Zhou, “Quantum entanglement concentration based on nonlinear optics for quantum communications,” Entropy 15, 1776–1820 (2013).
[CrossRef]

2012 (4)

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]

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2012).
[CrossRef]

M. Siomau, A. A. Kamli, S. A. Moiseev, and B. C. Sanders, “Entanglement creation with negative index metamaterials,” Phys. Rev. A 85, 050303(R) (2012).
[CrossRef]

2011 (3)

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

C. Wang, Y. S. Li, and L. Hao, “Optical implementation of quantum random walks using weak cross-Kerr media,” Chinese Sci. Bull. 56, 2088–2091 (2011).
[CrossRef]

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

2010 (4)

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]

P. P. Munhoz, J. A. Roversi, A. Vidiella-Barranco, and F. L. Semiao, “Bipartite quantum channels using multipartite cluster-type entangled coherent states,” Phys. Rev. A 81, 042305 (2010).
[CrossRef]

B. He, Y. Ren, and J. A. Bergou, “Universal entangler with photon pairs in arbitrary states,” J. Phys. B 43, 025502 (2010).
[CrossRef]

J. Gea-Banacloche, “Impossibility of large phase shifts via the giant Kerr effect with single-photon wave packets,” Phys. Rev. A 81, 043823 (2010).
[CrossRef]

2009 (6)

Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79, 022301 (2009).
[CrossRef]

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

B. He, M. Nadeem, and J. A. Bergou, “Scheme for generating coherent-state superpositions with realistic cross-Kerr nonlinearity,” Phys. Rev. A 79, 035802 (2009).
[CrossRef]

B. He, Y. Ren, and J. A. Bergou, “Creation of high-quality long-distance entanglement with flexible resources,” Phys. Rev. A 79, 052323 (2009).
[CrossRef]

N. B. An and J. Kim, “Cluster-type entangled coherent states: generation and application,” Phys. Rev. A 80, 042316 (2009).
[CrossRef]

M.-Y. Chen, M. W. Y. Tu, and W.-M. Zhang, “Entangling two superconducting LC coherent modes via a superconducting flux qubit,” Phys. Rev. B 80, 214538 (2009).
[CrossRef]

2008 (3)

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]

X.-H. Bao, Y. Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef]

2007 (4)

J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and quantum computation,” New J. Phys. 9, 16 (2007).
[CrossRef]

H. Prakash, N. Chandra, R. Prakash, and Shivani, “Improving the teleportation of entangled coherent states,” Phys. Rev. A 75, 044305 (2007).
[CrossRef]

Y. Guo and L. M. Kuang, “Near-deterministic generation of four-mode W-type entangled coherent states,” J. Phys. B 40, 3309–3318 (2007).
[CrossRef]

L. M. Kuang, Z. B. Chen, and J. W. Pan, “Generation of entangled coherent states for distant Bose–Einstein condensates via electromagnetically induced transparency,” Phys. Rev. A 76, 052324 (2007).
[CrossRef]

2006 (2)

H. Jeong and N. B. An, “Greenberger-Horne- Zeilinger-type and W-type entangled coherent states: generation and Bell-type inequality tests without photon counting,” Phys. Rev. A 74, 022104 (2006).
[CrossRef]

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

2005 (3)

B. Wang and L. M. Duan, “Engineering superpositions of coherent states in coherent optical pulses through cavity-assisted interaction,” Phys. Rev. A 72, 022320 (2005).
[CrossRef]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[CrossRef]

H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
[CrossRef]

2004 (5)

H. Jeong, M. S. Kim, T. C. Ralph, and B. S. Ham, “Generation of macroscopic superposition states with small nonlinearity,” Phys. Rev. A 70, 061801(R) (2004).
[CrossRef]

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

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent states with inefficient photon detection,” Phys. Rev. A 70, 020101 (2004).
[CrossRef]

S. Glancy, H. M. Vasconcelos, and T. C. Ralph, “Transmission of optical coherent-state qubits,” Phys. Rev. A 70, 022317 (2004).
[CrossRef]

N. B. An, “Optimal processing of quantum information via W-type entangled coherent states,” Phys. Rev. A 69, 022315 (2004).
[CrossRef]

2003 (6)

N. B. An, “Teleportation of coherent-state superpositions within a network,” Phys. Rev. A 68, 022321 (2003).
[CrossRef]

X. G. Wang, M. Feng, and B. C. Sanders, “Multipartite entangled states in coupled quantum dots and cavity QED,” Phys. Rev. A 67, 022302 (2003).
[CrossRef]

E. Solano, G. S. Agarwal, and H. Walther, “Strong driving- assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[CrossRef]

L. M. Kuang and L. Zhou, “Generation of atom-photon entangled states in atomic Bose-Einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2003).
[CrossRef]

S. J. van Enk, “Entanglement capabilities in infinite dimensions: multidimensional entangled coherent states,” Phys. Rev. Lett. 91, 017902 (2003).
[CrossRef]

D. Braje, V. Balić, G. Yin, and S. Harris, “Low-light-level non-linear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

2002 (3)

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Quantum dynamics of a cooper-pair box coupled to a micromechanical resonator,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef]

J. Clausen, L. Knöll, and D.-G. Welsch, “Lossy purification and detection of entangled coherent states,” Phys. Rev. A 66, 062303 (2002).
[CrossRef]

H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states,” Phys. Rev. A 65, 042305 (2002).
[CrossRef]

2001 (4)

S. J. van Enk and O. Hirota, “Entangled coherent states: teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
[CrossRef]

X. Wang, “Quantum teleportation of entangled coherent states,” Phys. Rev. A 64, 022302 (2001).
[CrossRef]

H. Jeong, M. S. Kim, and J. Lee, “Quantum information processing for a coherent superposition state via a mixed entangled coherent channel,” Phys. Rev. A 64, 052308 (2001).
[CrossRef]

H. Weinfurter and M. Zukowski, “Four-photon entanglement from down-conversion,” Phys. Rev. A 64, 010102(R) (2001).
[CrossRef]

2000 (1)

W. J. Munro, G. J. Milburn, and B. C. Sanders, “Entangled coherent-state qubits in an ion trap,” Phys. Rev. A 62, 052108 (2000).
[CrossRef]

1999 (3)

P. T. Cochrane, G. J. Milburn, and W. J. Munro, “Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping,” Phys. Rev. A 59, 2631–2634 (1999).
[CrossRef]

S. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[CrossRef]

C. C. Gerry, “Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium,” Phys. Rev. A 59, 4095–4098 (1999).
[CrossRef]

1998 (2)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[CrossRef]

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum nondemolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

1995 (2)

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

1994 (1)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[CrossRef]

1992 (1)

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811–6815 (1992).
[CrossRef]

1990 (1)

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132–5138 (1990).
[CrossRef]

1967 (1)

Y. Aharonov and L. Susskind, “Charge superselection rule,” Phys. Rev. 155, 1428–1431 (1967).
[CrossRef]

Agarwal, G. S.

E. Solano, G. S. Agarwal, and H. Walther, “Strong driving- assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[CrossRef]

Aharonov, Y.

Y. Aharonov and L. Susskind, “Charge superselection rule,” Phys. Rev. 155, 1428–1431 (1967).
[CrossRef]

An, N. B.

N. B. An and J. Kim, “Cluster-type entangled coherent states: generation and application,” Phys. Rev. A 80, 042316 (2009).
[CrossRef]

H. Jeong and N. B. An, “Greenberger-Horne- Zeilinger-type and W-type entangled coherent states: generation and Bell-type inequality tests without photon counting,” Phys. Rev. A 74, 022104 (2006).
[CrossRef]

N. B. An, “Optimal processing of quantum information via W-type entangled coherent states,” Phys. Rev. A 69, 022315 (2004).
[CrossRef]

N. B. An, “Teleportation of coherent-state superpositions within a network,” Phys. Rev. A 68, 022321 (2003).
[CrossRef]

Armour, A. D.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Quantum dynamics of a cooper-pair box coupled to a micromechanical resonator,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef]

Balic, V.

D. Braje, V. Balić, G. Yin, and S. Harris, “Low-light-level non-linear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Bao, X.-H.

X.-H. Bao, Y. Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[CrossRef]

Bergou, J. A.

B. He, Y. Ren, and J. A. Bergou, “Universal entangler with photon pairs in arbitrary states,” J. Phys. B 43, 025502 (2010).
[CrossRef]

B. He, Y. Ren, and J. A. Bergou, “Creation of high-quality long-distance entanglement with flexible resources,” Phys. Rev. A 79, 052323 (2009).
[CrossRef]

B. He, M. Nadeem, and J. A. Bergou, “Scheme for generating coherent-state superpositions with realistic cross-Kerr nonlinearity,” Phys. Rev. A 79, 035802 (2009).
[CrossRef]

Blencowe, M. P.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Quantum dynamics of a cooper-pair box coupled to a micromechanical resonator,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef]

Braje, D.

D. Braje, V. Balić, G. Yin, and S. Harris, “Low-light-level non-linear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Chandra, N.

H. Prakash, N. Chandra, R. Prakash, and Shivani, “Improving the teleportation of entangled coherent states,” Phys. Rev. A 75, 044305 (2007).
[CrossRef]

Chen, M.-Y.

M.-Y. Chen, M. W. Y. Tu, and W.-M. Zhang, “Entangling two superconducting LC coherent modes via a superconducting flux qubit,” Phys. Rev. B 80, 214538 (2009).
[CrossRef]

Chen, Z. B.

L. M. Kuang, Z. B. Chen, and J. W. Pan, “Generation of entangled coherent states for distant Bose–Einstein condensates via electromagnetically induced transparency,” Phys. Rev. A 76, 052324 (2007).
[CrossRef]

Chen, Z.-B.

X.-H. Bao, Y. Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

Chuang, I. L.

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

Clausen, J.

J. Clausen, L. Knöll, and D.-G. Welsch, “Lossy purification and detection of entangled coherent states,” Phys. Rev. A 66, 062303 (2002).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the generation of Bell-type ECSs with single photon. BS and PBS denote the beam splitter and polarization beam splitter, respectively. D1 and D2 denote the single-photon detectors.

Fig. 2.
Fig. 2.

Fidelity versus the phase shift θ of the obtained Bell-type ECS. The solid, dashed, and dot-dashed lines represent the cases in which the initial amplitude |α|=100 with Γ=0.5, |α|=100 with Γ=1, and |α|=200 with Γ=1, respectively, where Γ=γ/χ.

Fig. 3.
Fig. 3.

Coherence parameter C(t) versus the amplitude |α| of the initial state with |β|=3 and Γ=1.

Fig. 4.
Fig. 4.

Amplitude β versus the phase shift θ. The solid, dashed, and dot-dashed lines represent the cases in which the initial amplitude |α|=100 with Γ=0.5, |α|=100 with Γ=1, and |α|=200 with Γ=1, respectively.

Fig. 5.
Fig. 5.

Schematic of Bell-type ECS generation with entangled-photon pairs.

Fig. 6.
Fig. 6.

Concurrence of the obtained Bell-type ECSs versus the initial amplitude |α| with |β|=3 and Γ=1.

Fig. 7.
Fig. 7.

Fidelity of the obtained Bell-type ECSs versus phase shift θ.

Fig. 8.
Fig. 8.

Fidelity of the obtained Bell-type ECSs versus phase noise variance.

Equations (32)

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|CSS±=N±(|α±|α),
K|α|n=eiHt/|α|n=|αeinθ|n,
ρ(t)t=iχi,j[aiaiajaj,ρ(t)]+γ2i[2aiρ(t)aiaiaiρ(t)ρ(t)aiai],
|Ψin=12(|H+|V)|2α|2α.
|Ψ=12(|HU+|VL)|α,α,α,α,
|Ψ=12(|HU+|VL)|α,α,α,αK12(|HU|αeiθ,α,αeiθ,α+|VL|α,αeiθ,α,αeiθ),
|Ψ|α1,α2,α1,α2+|α1,α2,α1,α2,
|Ψ2,4|α2,α2+|α2,α2|α(eiθ1)2,α(eiθ1)2+|α(1eiθ)2,α(1eiθ)2.
Jiρ=γa^iρa^i,Liρ=γ2(a^ia^iρ+ρa^ia^i),
ρ(t)=limNk=1N1exp[i(Ji+Li)Δt+i,jKi,jΔt]ρ(t0),
ρ(t1)=12ei[(Ji+Li)Δt]ei,j[Ki,jΔt]{(|H+|V)(H|+V|)|αα||αα||αα||αα|}=12ei[(Ji+Li)Δt](|H|αeiθ1|α|αeiθ1|α+|V|α|αeiθ1|α|αeiθ1)(H|αeiθ1|α|αeiθ1|α|+V|α|αeiθ1|α|αeiθ1|)(|HH||A1αeiθ1,A1α,A1αeiθ1,A1αA1αeiθ1,A1α,A1αeiθ1,A1α|+|VV||A1α,A1αeiθ1,A1α,A1αeiθ1A1α,A1αeiθ1,A1α,A1αeiθ1|+C1|HV||A1αeiθ1,A1α,A1αeiθ1,A1αA1α,A1αeiθ1,A1α,A1αeiθ1|+C1|VH||A1α,A1αeiθ1,A1α,A1αeiθ1A1αeiθ1,A1α,A1αeiθ1,A1α|),
ρ(t)=limNk=1N1exp[i(Ji+Li)Δt+i,jKi,jΔt]ρ(t0)|HH||Aαeiθ,Aα,Aαeiθ,AαAαeiθ,Aα,Aαeiθ,Aα|+|VV||Aα,Aαeiθ,Aα,AαeiθAα,Aαeiθ,Aα,Aαeiθ|+C(t)|HV||Aαeiθ,Aα,Aαeiθ,AαAα,Aαeiθ,Aα,Aαeiθ|+C(t)|VH||Aα,Aαeiθ,Aα,AαeiθAαeiθ,Aα,Aαeiθ,Aα|,
C(t)=limNC1C2CN1=exp[4|α|2(χ2χ2+γ2eγt+γ2χ2+γ2eγtcosχtγχχ2+γ2eγtsinχt)].
ρ|λβ,β,λβ,βλβ,β,λβ,β|+|λβ,β,λβ,βλβ,β,λβ,β|+C(t)|λβ,β,λβ,βλβ,β,λβ,β|+C(t)|λβ,β,λβ,βλβ,β,λβ,β|,
ρ2,41+C(t)2N+2|Bell+α2,4Bell+α|+1C(t)2N2|Bellα2,4Bellα|,
|β|=2eγt/2|αsinχt2|.
F=1+C(t)2,
F=Ψ1|ρ2|Ψ1
|Ψin=12(|HH+|VV)|2α|2α.
|Ψ=12(|H,0,H,0+|0,V,0,V)|α,α,α,α.
|Ψ=12(|HH+|VV)|α,α,α,αK12(|HH|αeiθ,α,αeiθ,α+|VV|α,αeiθ,α,αeiθ).
ρ(t1)=12ei[(Ji+Li)Δt]ei,j[Ki,jΔt]{(|HH+|VV)(HH|+VV|)|αα||αα||αα||αα|}=12ei[(Ji+Li)Δt](|HH|αeiθ1|α|αeiθ1|α+|VV|α|αeiθ1|α|αeiθ1)(HH|αeiθ1|α|αeiθ1|α|+VV|α|αeiθ1|α|αeiθ1|)(|HHHH||A1αeiθ1,A1α,A1αeiθ1,A1αA1αeiθ1,A1α,A1αeiθ1,A1α|+|VVVV||A1α,A1αeiθ1,A1α,A1αeiθ1A1α,A1αeiθ1,A1α,A1αeiθ1|+C1|HHVV||A1αeiθ1,A1α,A1αeiθ1,A1αA1α,A1αeiθ1,A1α,A1αeiθ1|+C1|VVHH||A1α,A1αeiθ1,A1α,A1αeiθ1A1αeiθ1,A1α,A1αeiθ1,A1α|),
ρ(t)|HHHH||Aαeiθ,Aα,Aαeiθ,AαAαeiθ,Aα,Aαeiθ,Aα|+|VVVV||Aα,Aαeiθ,Aα,AαeiθAα,Aαeiθ,Aα,Aαeiθ|+C(t)|HHVV||Aαeiθ,Aα,Aαeiθ,AαAα,Aαeiθ,Aα,Aαeiθ|+C(t)|VVHH||Aα,Aαeiθ,Aα,AαeiθAαeiθ,Aα,Aαeiθ,Aα|,
C(ρ)=max{0,λ1λ2λ3λ4},
ρ2,4(1+C(t)N+4001+C(t)N+2N+201C(t)N+2N21C(t)N+2N2001C(t)N+2N21C(t)N+2N201+C(t)N+2N+2001+C(t)N4),
|ψS=α|0S+βdtϕ(t+th)|tS,
|ψP=|αaP,
a^Seiξ^S(th)eiμ^S(th)a^S,a^Peiξ^P(0)eiμ^P(0)a^P,
μ^K(t)κdτh(tτ)E^J(τ)E^J(τ),J,K{S,P},JK
ρout(α,β)=|α|2|0aS0||αeiξPaPαeiξP|+|β|2|1aS1||αei(ξP+θ)aPαei(ξP+θ)|+(α*βeiξS|1aS0|+αβ*eiξS|0aS1|)|αei(ξP+θ/2)aPαei(ξP+θ/2)|ξS,ξP,
ρ|eiξPλβ,eiξPβ,eiξPλβ,eiξPβeiξPλβ,eiξPβ,eiξPλβ,eiξPβ|+|eiξPλβ,eiξPβ,eiξPλβ,eiξPβeiξPλβ,eiξPβ,eiξPλβ,eiξPβ|+(eiξS+eiξS)|ei(ξP+θ2)2α,0,ei(ξP+θ2)2α,0ei(ξP+θ2)2α,0,ei(ξP+θ2)2α,0|ξS,ξP,
F=Bell+α|ρ2,4Tr(ρ2,4)|Bell+αe4θ2|α|2+2eθ2|α|2+4eθ2|α|22σS22(1+e2θ2|α|2)(1+e2σS2)+1/1+2θ2|α|2σP22(1+e2θ2|α|2)(1+e2σS2),σ+θ1.

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