Abstract

We propose two counterfactual schemes for tripartite entanglement distribution without any physical particles travelling through the quantum channel. One scheme arranges three participators to connect with the absorption object by using switch. Using the “chained” quantum Zeno effect, three participators can accomplish the task of entanglement distribution with unique counterfactual interference probability. Another scheme uses Michelson-type interferometer to swap two entanglement pairs such that the photons of three participators are entangled. Moreover, the distance of entanglement distribution is doubled as two distant absorption objects are used. We also discuss the implementation issues to show that the proposed schemes can be realized with current technology.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Counterfactual entanglement distribution without transmitting any particles

Qi Guo, Liu-Yong Cheng, Li Chen, Hong-Fu Wang, and Shou Zhang
Opt. Express 22(8) 8970-8984 (2014)

Counterfactual entanglement distribution using quantum dot spins

Yuanyuan Chen, Dong Jiang, Xuemei Gu, Ling Xie, and Lijun Chen
J. Opt. Soc. Am. B 33(4) 663-669 (2016)

Counterfactual entanglement swapping enables high-efficiency entanglement distribution

Qi Guo, Liu-Yong Cheng, Hong-Fu Wang, and Shou Zhang
Opt. Express 26(21) 27314-27325 (2018)

References

  • View by:
  • |
  • |
  • |

  1. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
    [Crossref]
  2. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
    [Crossref]
  3. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
    [Crossref]
  4. M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
    [Crossref] [PubMed]
  5. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
    [Crossref]
  6. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
    [Crossref] [PubMed]
  7. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
    [Crossref] [PubMed]
  8. A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
    [Crossref] [PubMed]
  9. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
    [Crossref] [PubMed]
  10. N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
    [Crossref] [PubMed]
  11. T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
    [Crossref] [PubMed]
  12. S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
    [Crossref]
  13. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891 (1998).
    [Crossref]
  14. G. Fauconnier, “Analogical counterfactuals,” In G. Fauconnier and E. Sweetser, eds., Spaces, Worlds, and Grammar, The University of Chicago Press, pp. 57–90 (1996).
  15. A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Found. Phys. 23, 987–997 (1993).
    [Crossref]
  16. O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
    [Crossref] [PubMed]
  17. T.-G. Noh, “Counterfactual quantum cryptography,” Phys. Rev. Lett. 103, 230501 (2009).
    [Crossref]
  18. H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
    [Crossref] [PubMed]
  19. Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual entanglement distribution without transmitting any particles,” Opt. Express 22, 8970–8984 (2014).
    [Crossref] [PubMed]
  20. L. Vaidman, “Impossibility of the counterfactual computation for all possible outcomes,” Phys. Rev. Lett. 98, 160403 (2007).
    [Crossref] [PubMed]
  21. L. Vaidman, “Comment on “Protocol for direct counterfactual quantum communication”,” Phys. Rev. Lett. 112, 208901 (2014).
    [Crossref]
  22. H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
    [Crossref]
  23. L.-M. Duan, M. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
    [Crossref] [PubMed]
  24. L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
    [Crossref]
  25. T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R. Hadfield, S. Miki, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, “Long-distance entanglement-based quantum key distribution over optical fiber,” Opt. Express 16, 19118–19126 (2008).
    [Crossref]
  26. H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
    [Crossref]
  27. C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
    [Crossref]
  28. C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
    [Crossref] [PubMed]
  29. Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

2014 (3)

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual entanglement distribution without transmitting any particles,” Opt. Express 22, 8970–8984 (2014).
[Crossref] [PubMed]

L. Vaidman, “Comment on “Protocol for direct counterfactual quantum communication”,” Phys. Rev. Lett. 112, 208901 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

2013 (2)

H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

2010 (1)

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

2009 (3)

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

T.-G. Noh, “Counterfactual quantum cryptography,” Phys. Rev. Lett. 103, 230501 (2009).
[Crossref]

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

2008 (2)

2007 (2)

L. Vaidman, “Impossibility of the counterfactual computation for all possible outcomes,” Phys. Rev. Lett. 98, 160403 (2007).
[Crossref] [PubMed]

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

2006 (2)

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
[Crossref]

2004 (2)

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

2003 (2)

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

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

2002 (1)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

2001 (1)

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

2000 (2)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

1998 (2)

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

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

1997 (1)

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

1993 (1)

A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Found. Phys. 23, 987–997 (1993).
[Crossref]

Acín, A.

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

Al-Amri, M.

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

Asobe, M.

Baek, B.

Barreiro, J. T.

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

Becher, C.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Benhelm, J.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Blatt, R.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Bonato, C.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Boschi, C. D. E.

L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
[Crossref]

Bouwmeester, D.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

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

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Bowen, W. P.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

Cao, Y.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Cao, Z.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Chen, L.

Chen, Y.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Cheng, L.-Y.

Cirac, J. I.

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

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

Cubitt, T. S.

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

Ding, D.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Duan, L.-M.

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

Dür, W.

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

Eibl, M.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Elitzur, A. C.

A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Found. Phys. 23, 987–997 (1993).
[Crossref]

Fauconnier, G.

G. Fauconnier, “Analogical counterfactuals,” In G. Fauconnier and E. Sweetser, eds., Spaces, Worlds, and Grammar, The University of Chicago Press, pp. 57–90 (1996).

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

Gudat, J.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Guo, Q.

Hadfield, R.

Häffner, H.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Hänsel, W.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Haupt, F.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Honjo, T.

Horodecki, K.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, M.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, P.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, R.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Hosten, O.

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

Hu, C.

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

Inoue, K.

James, D.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Jennewein, T.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

Jing, J.

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

Kamada, H.

Kiesel, N.

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

Körber, T.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Kwiat, P. G.

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

Lam, P. K.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Lancaster, G.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Lance, A. M.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Lewenstein, M.

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

Li, Y.-H.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Li, Z.-H.

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

Lukin, M.

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

Ma, X.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Mattle, K.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Miki, S.

Munro, W.

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

Nam, S. W.

Nishida, Y.

Noh, T.-G.

T.-G. Noh, “Counterfactual quantum cryptography,” Phys. Rev. Lett. 103, 230501 (2009).
[Crossref]

OBrien, J.

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

Oemrawsingh, S. S.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Pan, J.-W.

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

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Peng, C.-Z.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Peng, K.

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

Perseguers, S.

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

Peters, N. A.

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

Rakher, M. T.

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

Rarity, J.

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

Riebe, M.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Roncaglia, M.

L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
[Crossref]

Roos, C.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Salih, H.

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

Sanders, B. C.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Sasaki, M.

Schmid, C.

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

Schmidt-Kaler, F.

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

Shenoy, H. A.

H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
[Crossref]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

Solano, E.

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

Srikanth, R.

H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
[Crossref]

Srinivas, T.

H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
[Crossref]

Symul, T.

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

Tadanaga, O.

Takesue, H.

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

Tóth, G.

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

Vaidman, L.

L. Vaidman, “Comment on “Protocol for direct counterfactual quantum communication”,” Phys. Rev. Lett. 112, 208901 (2014).
[Crossref]

L. Vaidman, “Impossibility of the counterfactual computation for all possible outcomes,” Phys. Rev. Lett. 98, 160403 (2007).
[Crossref] [PubMed]

A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Found. Phys. 23, 987–997 (1993).
[Crossref]

van Exter, M. P.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

Venuti, L. C.

L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
[Crossref]

Verstraete, F.

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

Wang, H.-F.

Wang, Z.

Wehr, J.

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

Weihs, G.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

Weinfurter, H.

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

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

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

Wootters, W. K.

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

Xie, C.

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

Yamamoto, Y.

Yan, Y.

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

Yin, J.

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

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

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

Zhang, J.

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

Zhang, Q.

Zhang, S.

Zhao, F.

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

Zoller, P.

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

Zubairy, M. S.

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

EPL-Europhys. Lett. (1)

H. A. Shenoy, R. Srikanth, and T. Srinivas, “Semi-counterfactual cryptography,” EPL-Europhys. Lett. 103, 60008 (2013).
[Crossref]

Found. Phys. (1)

A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Found. Phys. 23, 987–997 (1993).
[Crossref]

Nature (4)

O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature 439, 949–952 (2006).
[Crossref] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum tele-portation,” Nature 390, 575–579 (1997).
[Crossref]

M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004).
[Crossref] [PubMed]

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

Opt. Express (2)

Phys. Rev. A (2)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, “Entanglement distribution in pure-state quantum networks,” Phys. Rev. A 77, 022308 (2008).
[Crossref]

Phys. Rev. B (1)

C. Hu, W. Munro, J. OBrien, and J. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
[Crossref]

Phys. Rev. Lett. (15)

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. 104, 160503 (2010).
[Crossref] [PubMed]

L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, “Long-distance entanglement in spin systems,” Phys. Rev. Lett. 96, 247206 (2006).
[Crossref]

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

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

T.-G. Noh, “Counterfactual quantum cryptography,” Phys. Rev. Lett. 103, 230501 (2009).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013).
[Crossref] [PubMed]

L. Vaidman, “Impossibility of the counterfactual computation for all possible outcomes,” Phys. Rev. Lett. 98, 160403 (2007).
[Crossref] [PubMed]

L. Vaidman, “Comment on “Protocol for direct counterfactual quantum communication”,” Phys. Rev. Lett. 112, 208901 (2014).
[Crossref]

H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014).
[Crossref]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729 (2000).
[Crossref] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[Crossref] [PubMed]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, “Tripartite quantum state sharing,” Phys. Rev. Lett. 92, 177903 (2004).
[Crossref] [PubMed]

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

N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, “Experimental observation of four-photon entangled dicke state with high fidelity,” Phys. Rev. Lett. 98, 063604 (2007).
[Crossref] [PubMed]

T. S. Cubitt, F. Verstraete, W. Dür, and J. I. Cirac, “Separable states can be used to distribute entanglement,” Phys. Rev. Lett. 91, 037902 (2003).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Other (2)

G. Fauconnier, “Analogical counterfactuals,” In G. Fauconnier and E. Sweetser, eds., Spaces, Worlds, and Grammar, The University of Chicago Press, pp. 57–90 (1996).

Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication with single photons,” in “CLEO: QELS Fundamental Science,” (Optical Society of America, 2014), pp. FM4A–6.

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 (9)

Fig. 1
Fig. 1 Experimental setup of tripartite counterfactual entanglement distribution. (a) Three participators connect with David (the absorption object) by using switch K. S stands for single-photon source, C: optical circulator, DD: photon detector, C-PBS: polarizing beam splitter in the circular basis, FM: Faraday mirror, PS is the phase shifter used to perform the transformation |ϕ〉 →−|ϕ〉, CQZE represents the “chained” quantum Zeno effect. (b) Experimental setup of “chained” quantum Zeno effect. SM: switchable mirror, SPR: switchable polarization rotator, D3: photon detector, OD: optical delay.
Fig. 2
Fig. 2 The parameters X m 3, Y m 3, Y m 2 Z m, Y m Z m 2 and Z m 3 in Eq. (4) versus the different values of m and n. (a) X m 3 approaches one with the increase of m and does not change with n, (b) Y m 3 is close to zero, (c) Y m 2 Z m is also close to zero, (d) Y m Z m 2 is also close to zero, (e) Z m 3 approaches one with the increase of m and n
Fig. 3
Fig. 3 The fidelity of tripartite counterfactual entanglement distribution versus different values of outer cycles m and inner cycles n.
Fig. 4
Fig. 4 The fidelity of tripartite counterfactual entanglement distribution versus the error coefficient s.
Fig. 5
Fig. 5 Principle of long-distance tripartite counterfactual entanglement distribution. BSC is a 50:50 beam splitter, BSm is the unbalanced beam splitter with reflectivity cos2 θ and transmissivity sin2θ (θ = π/2m), while BSn is also the unbalanced beam splitter with reflectivity cos2ϑ and transmissivity sin2ϑ (ϑ = π/2n), DA3, DC2, DC3, DC4 and DB3 are photon detectors, M is a normal mirror. One optical pulse is sent from Charlie and is then split by the BSC. If the absorption objects are present, the superposition pass through the path a1 and b1, and the photons of Alice and Bob would go through path A2 and B2, respectively. In contrast, if the absorption objects are absent, the superposition pass through the path a2 and b2, and the photons of Alice and Bob would go through path A1 and B1, respectively. Then the superposition comes back to BSC (the Michelson-type interferometer).
Fig. 6
Fig. 6 Experimental setup of long-distance tripartite counterfactual entanglement distribution. S stands for the single-photon source, C is the optical circulator, DC2, DD and DE are photon detectors, C-PBS is the polarizing beam splitter in the circular basis, PS is the phase shifter used to perform the transformation |ϕ〉 → −|ϕ〉, FM is the Faraday mirror, CQZE represents the “chained” quantum Zeno effect.
Fig. 7
Fig. 7 Efficiency of long-distance tripartite counterfactual entanglement distribution versus different values of outer and inner cycles m and n.
Fig. 8
Fig. 8 The fidelity of long-distance tripartite counterfactual entanglement distribution versus different numbers of users.
Fig. 9
Fig. 9 The fidelity of long-distance tripartite counterfactual entanglement distribution versus the error coefficient s.

Equations (8)

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

| ϕ m = X m | 0 a | 0 0 + Y m | 0 a | 1 o + Z m | 1 a | 1 o 2
X m = X m 1 cos θ Y m = Y m 1 cos θ Z m 1 sin θ Z m = ( Y m 1 sin θ Z m 1 cos θ ) cos n ϑ
| ϕ m = 1 2 ( X m 2 | 0 a | 0 b | 0 o + Y m 2 | 0 a | 0 b | 1 o + Y m Z m | 1 a | 0 b | 1 o + Y m Z m | 0 a | 1 b | 1 o + Z m 2 | 1 a | 1 b | 1 o )
| ϕ m = 1 2 ( X m 3 | 0 a | 0 b | 0 c | 0 o + Y m 3 | 0 a | 0 b | 0 c | 1 o + Y m 2 Z m | 1 a | 0 b | 0 c | 1 o + Y m 2 Z m | 0 a | 1 b | 0 c | 1 o + Y m 2 Z m | 0 a | 0 b | 1 c | 1 o + Y m Z m 2 | 1 a | 1 b | 0 c | 1 o + Y m Z m 2 | 1 a | 0 b | 1 c | 1 o + Y m Z m 2 | 0 a | 1 b | 1 c | 1 o + Z m 3 | 1 a | 1 b | 1 c | 1 o )
| ϕ m 1 2 [ ( | 0 a | 0 b | 0 c + | 1 a | 1 b | 1 c ) | 0 o + ( | 0 a | 0 b | 0 c | 1 a | 1 b | 1 c ) | 1 o ]
| ψ = | 0 a | 1 b + | 1 a | 0 b 2
| ψ AC = 1 2 [ ( | 0 A | 0 C A + | 1 A | 1 C A ) | 0 o + ( | 0 A | 0 C A | 1 A | 1 C A ) | 1 o ] | ψ BC = 1 2 [ ( | 0 B | 0 C B + | 1 B | 1 C B ) | 0 p + ( | 0 B | 0 C B | 1 B | 1 C B ) | 1 p ]
| ψ = 1 2 [ ( | 0 A | 1 B | 0 C + | 1 A | 0 B | 1 C ) | 0 o | 0 p + ( | 0 A | 1 B | 0 C | 1 A | 0 B | 1 C ) | 0 o | 1 p + ( | 0 A | 1 B | 0 C | 1 A | 0 B | 1 C ) | 1 o | 0 p + ( | 0 A | 1 B | 0 C + | 1 A | 0 B | 1 C ) | 1 o | 1 p ] = 1 2 [ ( | 0 A | 1 B | 0 C + | 1 A | 0 B | 1 C ) ( | 0 o | 0 p + | 1 o | 1 p ) + ( | 0 A | 1 B | 0 C | 1 A | 0 B | 1 C ) ( | 0 o | 1 p + | 1 o | 0 p ) ]

Metrics