Abstract

We investigate a candidate for the potential implementation of entanglement purification of two entangled nitrogen vacancy (N-V) centers that are coupled with two microtoroidal resonators. In our proposed entanglement purification protocol (EPP), mixed entangled ensembles can be purified with the local parity-check operations. With current and near-future technology, we can achieve the maximally entangled N-V centers nonlocally. And our EPP can be used in quantum repeaters for long-distance quantum communication.

© 2012 Optical Society of America

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    [CrossRef]
  8. B. Zhao, Z. B. Chen, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
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    [CrossRef]
  32. Q. Chen, W. L. Yang, M. Fang, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
    [CrossRef]
  33. P. B. Li, S. Y. Gao, and F. L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83, 054306 (2011).
    [CrossRef]
  34. L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, “Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae,” Science 326, 267–272 (2009).
    [CrossRef]
  35. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
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    [CrossRef]
  38. Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
    [CrossRef]
  39. P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
    [CrossRef]
  40. D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vuckovic, H. Park, and M. D. Lukin, “Deterministic coupling of a single nitrogen vacancy center to a photonic crystal cavity,” Nano Lett. 10, 3922–3926 (2010).
    [CrossRef]
  41. P. E. Barclay, C. Santori C, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009).
    [CrossRef]
  42. P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008).
    [CrossRef]
  43. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
    [CrossRef]

2012

T. J. Wang, S. Y. Song, and G. L. Long, “Quantum repeater based on spatial entanglement of photons and quantum-dot spins in optical microcavities,” Phys. Rev. A 85, 062311 (2012).
[CrossRef]

Y. B. Sheng, G. L. Long, and F. G. Deng, “One-step deterministic multipartite entanglement purification with linear optics,” Phys. Lett. A 376, 314C319 (2012).
[CrossRef]

D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Measurement and control of single nitrogen-vacancy center spins above 600 K,” Phys. Rev. X 2, 031001 (2012).
[CrossRef]

2011

Q. Chen, W. L. Yang, M. Fang, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
[CrossRef]

P. B. Li, S. Y. Gao, and F. L. Li, “Quantum-information transfer with nitrogen-vacancy centers coupled to a whispering-gallery microresonator,” Phys. Rev. A 83, 054306 (2011).
[CrossRef]

W. L. Yang, Z. Q. Yin, Y. Hu, M. Feng, and J. F. Du, “High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation,” Phys. Rev. A 84, 010301(R) (2011).

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

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011).
[CrossRef]

B. B. Zhang and Y. Q. Xu, “Atomic-ensemble-based quantum repeater against general polarization and phase noise,” Phys. Rev. A 84, 014304 (2011).
[CrossRef]

2010

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, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010).
[CrossRef]

W. L. Yang, Z. Y. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. 12, 113039 (2010).
[CrossRef]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vuckovic, H. Park, and M. D. Lukin, “Deterministic coupling of a single nitrogen vacancy center to a photonic crystal cavity,” Nano Lett. 10, 3922–3926 (2010).
[CrossRef]

2009

P. E. Barclay, C. Santori C, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009).
[CrossRef]

P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
[CrossRef]

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
[CrossRef]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55, 235–242 (2009).
[CrossRef]

L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, “Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae,” Science 326, 267–272 (2009).
[CrossRef]

F. Y. Hong and S. J. Xiong, “Robust quantum repeater with atomic ensembles and single-photon sources,” Phys. Rev. A 79, 052341 (2009).
[CrossRef]

Z. Q. Yin, Y. B. Zhao, Y. Yang, Z. -F. Han, and G. C. Guo, “Quantum repeaters free of polarization disturbance and phase noise,” Phys. Rev. A 79, 044302 (2009).
[CrossRef]

M. Gao, L. M. Liang, C. Z. Li, and X. B. Wang, “Robust quantum repeater with atomic ensembles against phase and polarization instability,” Phys. Rev. A 79, 042301 (2009).
[CrossRef]

2008

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008).
[CrossRef]

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

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008).
[CrossRef]

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[CrossRef]

P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008).
[CrossRef]

2007

Z. B. Chen, B. Zhao, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z. B. Chen, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef]

2006

Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[CrossRef]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

2003

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[CrossRef]

F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (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]

2001

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef]

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

1998

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

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

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]

1991

A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef]

Achard, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
[CrossRef]

Alkauskas, A.

D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Measurement and control of single nitrogen-vacancy center spins above 600 K,” Phys. Rev. X 2, 031001 (2012).
[CrossRef]

Aoki, T.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[CrossRef]

Awschalom, D. D.

D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Measurement and control of single nitrogen-vacancy center spins above 600 K,” Phys. Rev. X 2, 031001 (2012).
[CrossRef]

Balasubramanian, G.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
[CrossRef]

Barclay, P. E.

P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
[CrossRef]

P. E. Barclay, C. Santori C, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009).
[CrossRef]

Beausoleil, R. G.

P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
[CrossRef]

P. E. Barclay, C. Santori C, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009).
[CrossRef]

Beck, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
[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, 722–725 (1996).
[CrossRef]

Brassard, G.

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

Briegel, H. J.

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Brukner, C.

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

Buckley, B. B.

D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Measurement and control of single nitrogen-vacancy center spins above 600 K,” Phys. Rev. X 2, 031001 (2012).
[CrossRef]

Chen, Q.

Q. Chen, W. L. Yang, M. Fang, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
[CrossRef]

Chen, Y. -A.

Z. B. Chen, B. Zhao, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z. B. Chen, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef]

Chen, Z. B.

Z. B. Chen, B. Zhao, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Fault-tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z. B. Chen, Y. -A. Chen, J. Schmiedmayer, and J. W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef]

Childress, L.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

Christle, D. J.

D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, “Measurement and control of single nitrogen-vacancy center spins above 600 K,” Phys. Rev. X 2, 031001 (2012).
[CrossRef]

Chu, Y.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

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,” Nature 414, 413–418 (2001).
[CrossRef]

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Cook, A. K.

Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[CrossRef]

Cory, D. G.

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Figures (5)

Fig. 1.
Fig. 1.

Energy-level structure of the N-V center coupling to microtoroidal cavity. Quantum information is encoded in the spin-triplet state |ms=±1.

Fig. 2.
Fig. 2.

Schematic diagram showing the principle of PCG based on N-V centers.

Fig. 3.
Fig. 3.

Schematic diagram showing the principle of EPP process for entangled N-V centers. PCG represents the PCG operations performed by Alice and Bob, respectively. They also perform single-particle measurements locally after PCGs.

Fig. 4.
Fig. 4.

Fidelity after EPP based on experimental parameters. The cavity is assumed to be ideal with no cavity leakage κs/κ=0, and the decay rate of the N-V center is γ/2κ=0.05.

Fig. 5.
Fig. 5.

Entanglement fidelity of the iterated purification processes in practical conditions. The dotted line, the dot-dashed line, and the dashed line represent the final fidelity of the entanglement after performing the purification one time, two times, and three times, respectively. Finally, the thick solid line represents the entanglement fidelity after the purification process is iterated four times. The cavity is assumed to be ideal with no cavity leakage κs/κ=0, and the decay rate of the N-V center is γ/2κ=0.05. The cavity and N-V center coupling strength is g/κ=2.

Equations (14)

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ρt=iΔω[σz,ρ]iΩ2[σ+σ,ρ]+γ2(2σρσ+ρσ+σσ+σρ).
da^dt=[i(ωcωp)+κ+κs2]a^gσκa^in+H^,
dσdt=[i(ω0ωp)+γ2]σgσza^+G^.
r(ω)=[i(ω0ωp)+γ2][i(ωcωp)+κs2]+g2[i(ω0ωp)+γ2][i(ωcωp)+κ+κs2]+g2.
r0(ω)=i(ωcωp)+κs2i(ωcωp)+κs2+κ.
p|±1|±1|L+|R2|±1|±1e2iϕ|L+e2iϕ0|R2
p|±1|1|L+|R2|±1|1eiϕ+iϕ0(|L+|R)2.
|ψ+=12(|+1A|1B+|1A|+1B),
|ϕ+=12(|+1A|+1B+|1A|1B).
ρ=F|ψ+ψ+|+(1F)|ϕ+ϕ+|,
a1,a2b1,b2|ψ+a1,b1|ψ+a2,b212(|+1a1|+1a2|1b1|1b2|L|R2|L|R2+|1a1|1a2|+1b1|+1b2|L|R2|L|R2)+12(|+1a1|1a2|1b1|+1b2|L+|R2|L+|R2+|1a1|+1a2|+1b1|1b2|L+|R2|L+|R2).
a1,a2b1,b2|ϕ+a1,b1|ϕ+a2,b212(|+1a1|+1a2|+1b1|+1b2|L|R2|L|R2+|1a1|1a2|1b1|1b2|L|R2|L|R2)+12(|+1a1|1a2|+1b1|1b2|L+|R2|L+|R2+|1a1|+1a2|1b1|+1b2|L+|R2|L+|R2).
a1,a2b1,b2|ψ+a1,b1|ϕ+a2,b212(|+1a1|+1a2|1b1|+1b2|L|R2|L+|R2+|1a1|1a2|+1b1|1b2)|L|R2|L+|R2)+12(|+1a1|1a2|1b1|1b2|L+|R2|L|R2+|1a1|+1a2|+1b1|+1b2|L+|R2|L|R2),
a1,a2b1,b2|ϕ+a1,b1|ψ+a2,b212(|+1a1|+1a2|+1b1|1b2|L|R2|L+|R2+|1a1|1a2|1b1|+1b2|L|R2|L+|R2)+12(|+1a1|1a2|+1b1|+1b2|L+|R2|L|R2+|1a1|+1a2|1b1|1b2|L+|R2|L|R2).

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