Abstract

Entanglement purification is an indispensable ingredient in extended quantum communication networks and usually determines the efficiency and communication rate of quantum communication protocols. Different from all existing entanglement purification protocols (EPPs) where two or more copies of low quality mixed entangled states are selected from the same ensemble, here we describe a general and optimal EPP for arbitrary initial mixed states from different ensembles. We show that the successful operation of EPP may not obtain a higher fidelity mixed state, while the discarded source pair, which is usually regarded as a failure in existing EPPs, may have residual entanglement and can be reused to increase the yield of entanglement purification. We give the criterions of both the successful purification to obtain a higher fidelity mixed state and the existence of residual entanglement. Moreover, we reveal that entanglement purification procedure causes some entanglement loss. Finally, we provide an optimal approach to reduce the entanglement loss. This approach can also be used to increase the yield of entanglement purification. Our EPP may have potential application in long-distance quantum communications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Efficient hyperentanglement purification for three-photon systems with the fidelity-robust quantum gates and hyperentanglement link

Fang-Fang Du, Yong-Ting Liu, Zhen-Rong Shi, Yu-Xi Liang, Jun Tang, and Jun Liu
Opt. Express 27(19) 27046-27061 (2019)

Efficient entanglement purification of separate nitrogen-vacancy centers via coupling to microtoroidal resonators

Chuan Wang, Yong Zhang, Guang-Sheng Jin, and Ru Zhang
J. Opt. Soc. Am. B 29(12) 3349-3354 (2012)

Hyperentanglement purification using imperfect spatial entanglement

Tie-Jun Wang, Si-Chen Mi, and Chuan Wang
Opt. Express 25(3) 2969-2982 (2017)

References

  • View by:
  • |
  • |
  • |

  1. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
    [Crossref]
  2. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
    [Crossref]
  3. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
    [Crossref]
  4. G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
    [Crossref]
  5. F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
    [Crossref]
  6. W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
    [Crossref]
  7. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
    [Crossref]
  8. 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(13), 2818–2821 (1996).
    [Crossref]
  9. M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
    [Crossref]
  10. J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
    [Crossref]
  11. W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
    [Crossref]
  12. W. Dür and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
    [Crossref]
  13. Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
    [Crossref]
  14. 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(4), 042308 (2008).
    [Crossref]
  15. 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(3), 032307 (2011).
    [Crossref]
  16. C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19(25), 25685 (2011).
    [Crossref]
  17. D. Gonta and P. van Loock, “High-fidelity entanglement purification using chains of atoms and optical cavities,” Phys. Rev. A 86(5), 052312 (2012).
    [Crossref]
  18. Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
    [Crossref]
  19. B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90(5), 052309 (2014).
    [Crossref]
  20. G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
    [Crossref]
  21. G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
    [Crossref]
  22. T. J. Wang, S. C. Mi, and C. Wang, “Hyperentanglement purification using imperfect spatial entanglement,” Opt. Express 25(3), 2969 (2017).
    [Crossref]
  23. F. F. Du, Y. T. Liu, Z. R. Shi, Y. X. Liang, J. Tang, and J. Liu, “Efficient hyperentanglement purification for three-photon systems with the fidelity-robust quantum gates and hyperentanglement link,” Opt. Express 27(19), 27046 (2019).
    [Crossref]
  24. L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
    [Crossref]
  25. L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Geenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
    [Crossref]
  26. H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
    [Crossref]
  27. J. Miguel-Ramiro and W. Dür, “Efficient entanglement purification protocols for d-level systems,” Phys. Rev. A 98(4), 042309 (2018).
    [Crossref]
  28. F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
    [Crossref]
  29. S. Krastanov, V. V. Albert, and L. Jiang, “Optimized entanglement purification,” Quantum 3, 123123 (2019).
    [Crossref]
  30. J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
    [Crossref]
  31. L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
    [Crossref]
  32. R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
    [Crossref]
  33. C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
    [Crossref]
  34. Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
    [Crossref]
  35. Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
    [Crossref]
  36. X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044304 (2010).
    [Crossref]
  37. Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5(1), 7815 (2015).
    [Crossref]
  38. M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
    [Crossref]
  39. M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
    [Crossref]
  40. M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
    [Crossref]
  41. J. Wallnöfer and W. Dür, “Measurement-based quantum communication with resource states generated by entanglement purification,” Phys. Rev. A 95(1), 012303 (2017).
    [Crossref]
  42. 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(26), 5932–5935 (1998).
    [Crossref]
  43. O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
    [Crossref]
  44. M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
    [Crossref]
  45. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
    [Crossref]
  46. C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
    [Crossref]
  47. S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78(26), 5022–5025 (1997).
    [Crossref]

2019 (2)

2018 (4)

J. Miguel-Ramiro and W. Dür, “Efficient entanglement purification protocols for d-level systems,” Phys. Rev. A 98(4), 042309 (2018).
[Crossref]

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

2017 (6)

J. Wallnöfer and W. Dür, “Measurement-based quantum communication with resource states generated by entanglement purification,” Phys. Rev. A 95(1), 012303 (2017).
[Crossref]

T. J. Wang, S. C. Mi, and C. Wang, “Hyperentanglement purification using imperfect spatial entanglement,” Opt. Express 25(3), 2969 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Geenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
[Crossref]

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

2016 (2)

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
[Crossref]

2015 (1)

Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5(1), 7815 (2015).
[Crossref]

2014 (2)

M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
[Crossref]

B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90(5), 052309 (2014).
[Crossref]

2013 (2)

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
[Crossref]

2012 (2)

M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
[Crossref]

D. Gonta and P. van Loock, “High-fidelity entanglement purification using chains of atoms and optical cavities,” Phys. Rev. A 86(5), 052312 (2012).
[Crossref]

2011 (2)

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(3), 032307 (2011).
[Crossref]

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19(25), 25685 (2011).
[Crossref]

2010 (3)

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
[Crossref]

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044304 (2010).
[Crossref]

2008 (1)

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(4), 042308 (2008).
[Crossref]

2007 (2)

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[Crossref]

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

2006 (1)

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

2003 (4)

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

W. Dür and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

2002 (2)

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
[Crossref]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
[Crossref]

2001 (1)

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

1999 (1)

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

1998 (3)

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[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(26), 5932–5935 (1998).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
[Crossref]

1997 (1)

S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78(26), 5022–5025 (1997).
[Crossref]

1996 (3)

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

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(13), 2818–2821 (1996).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

Ai, Q.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

Albert, V. V.

S. Krastanov, V. V. Albert, and L. Jiang, “Optimized entanglement purification,” Quantum 3, 123123 (2019).
[Crossref]

Alsaedi, A.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

Aschauer, H.

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

Benjamin, S.

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

Bennett, C. H.

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Bernstein, H. J.

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

Berthiaume, A.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

Blakestad, R. B.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Briegel, H. J.

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
[Crossref]

M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
[Crossref]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

W. Dür and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[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(26), 5932–5935 (1998).
[Crossref]

Britton, J.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Brukner, C.

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Bužek, V.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

Chen, L. K.

L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
[Crossref]

Cheong, Y. W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[Crossref]

Cirac, J. I.

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(26), 5932–5935 (1998).
[Crossref]

Collins, O. A.

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Deng, F. G.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
[Crossref]

B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90(5), 052309 (2014).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
[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(4), 042308 (2008).
[Crossref]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

Deutsch, D.

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(13), 2818–2821 (1996).
[Crossref]

Ding, D. S.

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

Doherty, A. C.

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Du, F. F.

Dunjko, V.

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

Dür, W.

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

J. Miguel-Ramiro and W. Dür, “Efficient entanglement purification protocols for d-level systems,” Phys. Rev. A 98(4), 042309 (2018).
[Crossref]

J. Wallnöfer and W. Dür, “Measurement-based quantum communication with resource states generated by entanglement purification,” Phys. Rev. A 95(1), 012303 (2017).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
[Crossref]

M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
[Crossref]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

W. Dür and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[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(26), 5932–5935 (1998).
[Crossref]

Ekert, A.

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(13), 2818–2821 (1996).
[Crossref]

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

Elkouss, D.

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Gasparoni, S.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

Gonta, D.

D. Gonta and P. van Loock, “High-fidelity entanglement purification using chains of atoms and optical cavities,” Phys. Rev. A 86(5), 052312 (2012).
[Crossref]

Guo, G. C.

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

Hayat, T.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

Hill, S.

S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78(26), 5022–5025 (1997).
[Crossref]

Hillery, M.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

Jenkins, S. D.

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

Jiang, L.

S. Krastanov, V. V. Albert, and L. Jiang, “Optimized entanglement purification,” Quantum 3, 123123 (2019).
[Crossref]

Jin, G. S.

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(3), 032307 (2011).
[Crossref]

Jost, J. D.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Jozsa, R.

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(13), 2818–2821 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Kennedy, T. A. B.

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

Knight, P. L.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

Knill, E.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Krastanov, S.

S. Krastanov, V. V. Albert, and L. Jiang, “Optimized entanglement purification,” Quantum 3, 123123 (2019).
[Crossref]

Kuzmich, A.

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

Langer, C.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Lee, H. W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[Crossref]

Lee, J.

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[Crossref]

Lee, S. W.

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[Crossref]

Leibfried, D.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Li, T.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

Li, X. H.

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044304 (2010).
[Crossref]

Liang, Y. X.

Liu, J.

Liu, Q.

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
[Crossref]

Liu, X. S.

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
[Crossref]

Liu, Y. T.

Long, G. L.

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
[Crossref]

Macchiavello, C.

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(13), 2818–2821 (1996).
[Crossref]

Mi, S. C.

Miguel-Ramiro, J.

J. Miguel-Ramiro and W. Dür, “Efficient entanglement purification protocols for d-level systems,” Phys. Rev. A 98(4), 042309 (2018).
[Crossref]

Murao, M.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

Ozeri, R.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Pan, J. W.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
[Crossref]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Peres, A.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Pirker, A.

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

Plenio, M. B.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

Popescu, S.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[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(13), 2818–2821 (1996).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

Reichle, R.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Ren, B. C.

B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90(5), 052309 (2014).
[Crossref]

Rozpeedek, F.

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Sanpera, A.

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(13), 2818–2821 (1996).
[Crossref]

Schiet, T.

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Schumacher, B.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

Seidelin, S.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Sheng, Y. B.

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Geenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5(1), 7815 (2015).
[Crossref]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
[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(4), 042308 (2008).
[Crossref]

Shi, B. S.

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

Shi, Z. R.

Simon, C.

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
[Crossref]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Smolin, J. A.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

Tang, J.

Ursin, R.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

van Loock, P.

D. Gonta and P. van Loock, “High-fidelity entanglement purification using chains of atoms and optical cavities,” Phys. Rev. A 86(5), 052312 (2012).
[Crossref]

Vedral, V.

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

Wallnöfer, J.

J. Wallnöfer and W. Dür, “Measurement-based quantum communication with resource states generated by entanglement purification,” Phys. Rev. A 95(1), 012303 (2017).
[Crossref]

Wang, C.

Wang, G. Y.

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
[Crossref]

Wang, T. J.

Wehner, S.

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Weihs, G.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

Wineland, D. J.

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
[Crossref]

S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78(26), 5022–5025 (1997).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

Xiong, J.

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

Xu, X. S.

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

Xu et al., P.

L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
[Crossref]

Yong, H. L.

L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
[Crossref]

Zeilinger, A.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Zhang, H.

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

Zhang, R.

Zhang, W.

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

Zhang, Y.

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(3), 032307 (2011).
[Crossref]

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19(25), 25685 (2011).
[Crossref]

Zhou, H. Y.

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(4), 042308 (2008).
[Crossref]

Zhou, L.

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Geenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5(1), 7815 (2015).
[Crossref]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

Zoller, P.

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(26), 5932–5935 (1998).
[Crossref]

Zwerger, M.

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
[Crossref]

M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
[Crossref]

Ann. Phys. (1)

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Geenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

Nat. Photonics (1)

L. K. Chen, H. L. Yong, and P. Xu et al., “Experimental nested purification for a linear optical quantum repeater,” Nat. Photonics 11(11), 695–699 (2017).
[Crossref]

Nature (3)

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423(6938), 417–422 (2003).
[Crossref]

J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Opt. Express (3)

Phys. Rev. A (21)

D. Gonta and P. van Loock, “High-fidelity entanglement purification using chains of atoms and optical cavities,” Phys. Rev. A 86(5), 052312 (2012).
[Crossref]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90(5), 052309 (2014).
[Crossref]

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94(3), 032319 (2016).
[Crossref]

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
[Crossref]

F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

H. Zhang, Q. Liu, X. S. Xu, J. Xiong, A. Alsaedi, T. Hayat, and F. G. Deng, “Polarization entanglement purification of nonlocal microwave photons based on the cross-kerr effect in circuit qed,” Phys. Rev. A 96(5), 052330 (2017).
[Crossref]

J. Miguel-Ramiro and W. Dür, “Efficient entanglement purification protocols for d-level systems,” Phys. Rev. A 98(4), 042309 (2018).
[Crossref]

F. Rozpeędek, T. Schiet, D. Elkouss, A. C. Doherty, and S. Wehner, “Optimizing practical entanglement distillation,” Phys. Rev. A 97(6), 062333 (2018).
[Crossref]

Y. W. Cheong, S. W. Lee, J. Lee, and H. W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A 76(4), 042314 (2007).
[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(4), 042308 (2008).
[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(3), 032307 (2011).
[Crossref]

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
[Crossref]

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044304 (2010).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Robustness of hashing protocols for entanglement purification,” Phys. Rev. A 90(1), 012314 (2014).
[Crossref]

J. Wallnöfer and W. Dür, “Measurement-based quantum communication with resource states generated by entanglement purification,” Phys. Rev. A 95(1), 012303 (2017).
[Crossref]

M. Zwerger, W. Dür, and H. J. Briegel, “Measurement-based quantum repeaters,” Phys. Rev. A 85(6), 062326 (2012).
[Crossref]

C. H. Bennett, H. J. Bernstein, S. Popescu, and S. Benjamin, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53(4), 2046–2052 (1996).
[Crossref]

Phys. Rev. Appl. (1)

G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018).
[Crossref]

Phys. Rev. Lett. (14)

S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78(26), 5022–5025 (1997).
[Crossref]

M. Zwerger, H. J. Briegel, and W. Dür, “Universal and optimal error thresholds for measurement-based entanglement purification,” Phys. Rev. Lett. 110(26), 260503 (2013).
[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(26), 5932–5935 (1998).
[Crossref]

O. A. Collins, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “Multiplexed memory-insensitive quantum repeaters,” Phys. Rev. Lett. 98(6), 060502 (2007).
[Crossref]

M. Zwerger, A. Pirker, V. Dunjko, H. J. Briegel, and W. Dür, “Long-range big quantum-data transmission,” Phys. Rev. Lett. 120(3), 030503 (2018).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998).
[Crossref]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
[Crossref]

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

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(13), 2818–2821 (1996).
[Crossref]

W. Dür, H. Aschauer, and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

W. Dür and H. J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett. 91(10), 107903 (2003).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

Quantum (1)

S. Krastanov, V. V. Albert, and L. Jiang, “Optimized entanglement purification,” Quantum 3, 123123 (2019).
[Crossref]

Sci. Rep. (2)

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5(1), 7815 (2015).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. The schematic diagram of entanglement purification for purifying the residual entanglement based on Ref. [7]. S$_{1}$ and S$_{2}$ are two different ideal sources. The QND represents the quantum nondemolition detection gate, i.e. the CNOT gate.
Fig. 2.
Fig. 2. The relationship between the entanglement fidelity $F''$ and $F_{1}$. We control $F_{2}=0.51$ and change $F_{1}$ from 0.52 to 0.99.
Fig. 3.
Fig. 3. The relationship between the fidelity $F'''$ and $F_{1}$. We control $F_{2}=0.51$ and change $F_{1}$ from 0.52 to 0.99.
Fig. 4.
Fig. 4. The relationship between the entanglement transformation efficiency $\eta$ and the fidelity $F_{2}$. Here we control $F_{1}=0.99$ and change $F_{2}$ from 0.5 to 0.99. Curve A and B correspond to our EPP and the EPP without the residual entanglement like Ref. [7,8,14], respectively.

Tables (1)

Tables Icon

Table 1. The relationship between the initial fidelity of the mixed states and the yield.

Equations (33)

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

ρ 1 = F 1 | ϕ + a 1 b 1 ϕ + | + ( 1 F 1 ) | ψ + a 1 b 1 ψ + | ,
ρ 2 = F 2 | ϕ + a 2 b 2 ϕ + | + ( 1 F 2 ) | ψ + a 2 b 2 ψ + | .
| ϕ + a 1 b 1 | ϕ + a 2 b 2 | ϕ + a 3 b 3 | ϕ + a 4 b 4 , | ϕ + a 1 b 1 | ψ + a 2 b 2 | ϕ + a 1 b 3 | ψ + a 4 b 4 , | ψ + a 1 b 1 | ϕ + a 2 b 2 | ψ + a 3 b 3 | ψ + a 4 b 4 , | ψ + a 1 b 1 | ψ + a 2 b 2 | ψ + a 3 b 3 | ϕ + a 4 b 4 .
F = F 1 F 2 F 1 F 2 + ( 1 F 1 ) ( 1 F 2 ) .
ρ = F | ϕ + a 3 b 3 ϕ + | + ( 1 F ) | ψ + a 3 b 3 ψ + | ,
F = F 1 ( 1 F 2 ) F 1 ( 1 F 2 ) + ( 1 F 1 ) F 2 .
F = F 2 F 2 + ( 1 F ) 2 = F 1 2 ( 1 F 2 ) 2 F 1 2 ( 1 F 2 ) 2 + F 2 2 ( 1 F 1 ) 2 .
F 1 1 F 1 > F 2 2 ( 1 F 2 ) 2 .
ρ 1 = F 1 | ϕ + a 1 b 1 ϕ + | + A 1 | ϕ a 1 b 1 ϕ | + B 1 | ψ + a 1 b 1 ψ + | + C 1 | ψ a 1 b 1 ψ | ,
ρ 2 = F 2 | ϕ + a 2 b 2 ϕ + | + A 2 | ϕ a 2 b 2 ϕ | + B 2 | ψ + a 2 b 2 ψ + | + C 2 | ψ a 2 b 2 ψ | .
ρ 3 = F 3 | ϕ + a 1 b 1 ϕ + | + A 3 | ϕ a 1 b 1 ϕ | + B 3 | ψ + a 1 b 1 ψ + | + C 3 | ψ a 1 b 1 ψ | ,
F 3 = F 1 F 2 + A 1 A 2 N 1 , A 3 = F 1 A 2 + A 1 F 2 N 1 , B 3 = B 1 B 2 + C 1 C 2 N 1 , C 3 = B 1 C 2 + C 1 B 2 N 1 .
F 2 1 2 A 2 > ( F 1 1 2 ) ( F 1 + A 1 ) F 1 [ 1 ( F 1 + A 1 ) ] .
ρ 4 = F 4 | ϕ + a 1 b 1 ϕ + | + A 4 | ϕ a 1 b 1 ϕ | + B 4 | ψ + a 1 b 1 ψ + | + C 4 | ψ a 1 b 1 ψ | ,
F 4 = F 1 B 2 + A 1 C 2 N 2 , A 4 = A 1 B 2 + F 1 C 2 N 2 , B 4 = B 1 F 2 + C 1 A 2 N 2 , C 4 = C 1 F 2 + B 1 A 2 N 2 .
B 2 C 2 0.5 C 2 > B 1 + C 1 0.5 A 1 , ( F 4 > 1 2 ) , C 2 B 2 0.5 B 2 > B 1 + C 1 0.5 A 1 , ( A 4 > 1 2 ) .
η = E ( ρ ) P 1 + E ( ρ ) P 2 E ( ρ 1 ) + E ( ρ 2 ) .
E ( | ψ ) = h ( 1 + 1 C ( | ψ ) 2 2 ) ,
h ( x ) = x l o g 2 x ( 1 x ) l o g 2 ( 1 x ) .
C ( | ψ ) = | ψ | ψ ~ | .
E ( ρ ) = m i n i p i E ( | ψ i ) .
E ( ρ ) = h ( 1 + 1 C ( ρ ) 2 2 ) ,
C ( ρ ) = m a x { 0 , λ 1 λ 2 λ 3 λ 4 } .
η = E ( ρ ) P 1 E ( ρ 1 ) + E ( ρ 2 ) ,
F 1 F 2 + A 1 A 2 ( F 1 + A 1 ) ( F 2 + A 2 ) + ( 1 F 1 A 1 ) ( 1 F 2 A 2 ) > F 1 .
F 1 [ 2 ( F 1 + A 1 ) ( F 2 + A 2 ) ( F 1 + A 1 ) ( F 2 + A 2 ) + 1 ] ( F 1 F 2 + A 1 A 2 ) < 0.
2 F 1 ( F 1 + A 1 1 ) F 2 + ( F 1 + A 1 ) ( 2 F 1 1 ) A 2 + F 1 [ 1 ( F 1 + A 1 ) ] < 0.
F 1 ( F 1 + A 1 1 ) ( 2 F 1 1 ) + ( F 1 + A 1 ) ( 2 F 1 1 ) A 2 < 0.
F 2 1 2 A 2 > ( F 1 1 2 ) ( F 1 + A 1 ) F 1 [ 1 ( F 1 + A 1 ) ] .
F 1 B 2 + A 1 C 2 ( F 1 + A 1 ) ( B 2 + C 2 ) + ( F 2 + A 2 ) ( B 1 + C 1 ) > 1 2 .
1 2 B 2 A 1 B 2 + A 1 C 2 > 1 2 ( B 1 + C 1 + C 2 ) B 1 C 2 C 1 C 2 .
( B 2 C 2 ) ( 1 2 A 1 ) > ( B 1 + C 1 ) ( 1 2 C 2 ) .
B 2 C 2 0.5 C 2 > B 1 + C 1 0.5 A 1 .

Metrics