Abstract

We propose a quasi-counterfactual quantum swap gate for exchanging Alice’s unknown photon state and Bob’s unknown atomic state under the condition that only Alice’s photon may appear in the transmission channel between Alice and Bob, while the probability of the existence of photon in the transmission channel is controllable and can tend to zero. Unlike standard counterfactual quantum communication protocols, quantum states exchange in present scenario is achieved by multiple phase operations, rather than multiple measurements. The total effect of those operations can be considered as a unitary time evolution operator. Therefore, the communication fidelity and efficiency of our protocol are always one if system imperfection and channel noise are not considered. Compared to standard counterfactual communication protocols, our protocol is easy to implement. We also show that it can be easily converted to a standard counterfactual one.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

2017 (3)

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

C. Liu, J. H. Liu, J. X. Zhang, and S.-Y. Zhu, “The experimental demonstration of high efficiency interaction-free measurement for quantum counterfactual-like communication,” Scientific Reports 7, 10875 (2017).
[Crossref] [PubMed]

2016 (3)

H. Salih, “Protocol for counterfactually transporting an unknown qubit,” Front. Phys. 3, 94 (2016).
[Crossref]

D. R. M. Arvidsson-Shukur and C. H. W. Barnes, “Quantum counterfactual communication without a weak trace,” Phys. Rev. A 94, 062303 (2016).
[Crossref]

Y. Y. Chen, D. Jiang, X. M. Gu, L. Xie, and L. J. Chen, “Counterfactual entanglement distribution using quantum dot spins,” J. Opt. Soc. Am. B 33, 663–669 (2016).
[Crossref]

2015 (4)

Y. Chen, X. Gu, D. Jiang, L. Xie, and L. Chen, “Tripartite counterfactual entanglement distribution,” Opt. Express 23, 21193–21203 (2015).
[Crossref] [PubMed]

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015).
[Crossref]

F. Li, J.-X. Zhang, and S.-Y. Zhu, “Numerical simulation of the effect of dissipation and phase fluctuation in a direct communication scheme,” J. Phys. B: At. Mol. Opt. Phys. 48, 115506 (2015).
[Crossref]

2014 (6)

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities,” Phys. Rev. A 90, 042327 (2014).
[Crossref]

H. Salih, “Tripartite counterfactual quantum cryptography,” Phys. Rev. A 90, 012333 (2014).
[Crossref]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct quantum communication with almost invisible photons,” Phys. Rev. A 89, 052334 (2014).
[Crossref]

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]

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

2013 (5)

A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[Crossref] [PubMed]

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
[Crossref] [PubMed]

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]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Comment on “Past of a quantum particle”,” Phys. Rev. A 88, 046102 (2013).
[Crossref]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref] [PubMed]

2009 (1)

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

2008 (1)

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

2007 (1)

M. Pavičić, “Nondestructive interaction-free atom-photon controlled-NOT gate,” Phys. Rev. A 75, 032342 (2007).
[Crossref]

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]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

2005 (1)

J. Cho and H.-W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett. 95, 160501 (2005).
[Crossref] [PubMed]

2004 (1)

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

2002 (1)

K. Boströem and T. Felbinger, “Deterministic secure direct communication using entanglement,” Phys. Rev. Lett. 89, 187902 (2002).
[Crossref]

2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

1999 (1)

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

1995 (1)

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

1993 (1)

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

Agnew, M.

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
[Crossref] [PubMed]

Al-Amri, M.

Z.-H. Li, M. S. Zubairy, and M. Al-Amri, “Quantum secure group communication,” Scientific Reports 8, 3899 (2018).
[Crossref] [PubMed]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015).
[Crossref]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct quantum communication with almost invisible photons,” Phys. Rev. A 89, 052334 (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]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Comment on “Past of a quantum particle”,” Phys. Rev. A 88, 046102 (2013).
[Crossref]

Arvidsson-Shukur, D. R. M.

D. R. M. Arvidsson-Shukur and C. H. W. Barnes, “Quantum counterfactual communication without a weak trace,” Phys. Rev. A 94, 062303 (2016).
[Crossref]

Barnes, C. H. W.

D. R. M. Arvidsson-Shukur and C. H. W. Barnes, “Quantum counterfactual communication without a weak trace,” Phys. Rev. A 94, 062303 (2016).
[Crossref]

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]

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Birnbaum, K. M.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Boca, A.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Boozer, A. D.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Boströem, K.

K. Boströem and T. Felbinger, “Deterministic secure direct communication using entanglement,” Phys. Rev. Lett. 89, 187902 (2002).
[Crossref]

Cao, Y.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Cao, Z.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Chen, L.

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

Y. Chen, X. Gu, D. Jiang, L. Xie, and L. Chen, “Tripartite counterfactual entanglement distribution,” Opt. Express 23, 21193–21203 (2015).
[Crossref] [PubMed]

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]

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities,” Phys. Rev. A 90, 042327 (2014).
[Crossref]

Chen, L. J.

Chen, T. Y.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Chen, Y.

Chen, Y. A.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Chen, Y. Y.

Cheng, L.Y.

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

Cheng, L.-Y.

Q. Guo, L.-Y. Cheng, H.-F. Wang, and S. Zhang, “Counterfactual entanglement swapping enables high-efficiency entanglement distribution,” Opt. Express 26, 27314–27325 (2018).
[Crossref] [PubMed]

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities,” Phys. Rev. A 90, 042327 (2014).
[Crossref]

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]

Cho, J.

J. Cho and H.-W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett. 95, 160501 (2005).
[Crossref] [PubMed]

Donohue, J. M.

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
[Crossref] [PubMed]

Elitzur, A. C.

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

Felbinger, T.

K. Boströem and T. Felbinger, “Deterministic secure direct communication using entanglement,” Phys. Rev. Lett. 89, 187902 (2002).
[Crossref]

Fong, K. Y.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

Gu, X.

Gu, X. M.

Guo, Q.

Q. Guo, L.-Y. Cheng, H.-F. Wang, and S. Zhang, “Counterfactual entanglement swapping enables high-efficiency entanglement distribution,” Opt. Express 26, 27314–27325 (2018).
[Crossref] [PubMed]

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

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]

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities,” Phys. Rev. A 90, 042327 (2014).
[Crossref]

Guo, X.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

Henage, T.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Herzog, T.

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

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]

Isenhower, L.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

Jeong, Y.

F. Zaman, Y. Jeong, and H. Shin, “Counterfactual Bell-state analysis,” Scientific Reports 8, 14641 (2018).
[Crossref] [PubMed]

Jiang, D.

Jiang, L.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

Johnson, T. A.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Kalb, N.

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

Kasevich, M. A.

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

Kimble, H. J.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Kulatunga, P. B.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[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]

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

Lavoie, J.

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
[Crossref] [PubMed]

Lee, H.-W.

J. Cho and H.-W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett. 95, 160501 (2005).
[Crossref] [PubMed]

Li, F.

F. Li, J.-X. Zhang, and S.-Y. Zhu, “Numerical simulation of the effect of dissipation and phase fluctuation in a direct communication scheme,” J. Phys. B: At. Mol. Opt. Phys. 48, 115506 (2015).
[Crossref]

Li, Y. H.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Li, Z.-H.

Z.-H. Li, M. S. Zubairy, and M. Al-Amri, “Quantum secure group communication,” Scientific Reports 8, 3899 (2018).
[Crossref] [PubMed]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015).
[Crossref]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct quantum communication with almost invisible photons,” Phys. Rev. A 89, 052334 (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]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Comment on “Past of a quantum particle”,” Phys. Rev. A 88, 046102 (2013).
[Crossref]

Liu, C.

C. Liu, J. -H. Liu, J.-X. Zhang, and S. -Y. Zhu, “Improvement of reliability in multi interferometer-based counterfactual deterministic communication with dissipation compensation,” Opt. Express 26, 2261–2269 (2018).
[Crossref] [PubMed]

C. Liu, J. H. Liu, J. X. Zhang, and S.-Y. Zhu, “The experimental demonstration of high efficiency interaction-free measurement for quantum counterfactual-like communication,” Scientific Reports 7, 10875 (2017).
[Crossref] [PubMed]

Liu, J. H.

C. Liu, J. H. Liu, J. X. Zhang, and S.-Y. Zhu, “The experimental demonstration of high efficiency interaction-free measurement for quantum counterfactual-like communication,” Scientific Reports 7, 10875 (2017).
[Crossref] [PubMed]

Liu, J. -H.

Ma, X. F.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Ma, X.-S.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

McKeever, J.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Miller, R.

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum Rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004).
[Crossref] [PubMed]

Mitchell, J. R.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

Nairz, O.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

Nölleke, C.

A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[Crossref] [PubMed]

Pan, J. W.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Pavicic, M.

M. Pavičić, “Nondestructive interaction-free atom-photon controlled-NOT gate,” Phys. Rev. A 75, 032342 (2007).
[Crossref]

Peng, C. Z.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

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]

Proite, N.

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (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]

Reiserer, A.

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref] [PubMed]

A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[Crossref] [PubMed]

Rempe, G.

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref] [PubMed]

A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[Crossref] [PubMed]

Resch, K. J.

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
[Crossref] [PubMed]

Ritter, S.

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref] [PubMed]

A. Reiserer, C. Nölleke, S. Ritter, and G. Rempe, “Ground-state cooling of a single atom at the center of an optical cavity,” Phys. Rev. Lett. 110, 223003 (2013).
[Crossref] [PubMed]

Saffman, M.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Salih, H.

H. Salih, “Protocol for counterfactually transporting an unknown qubit,” Front. Phys. 3, 94 (2016).
[Crossref]

H. Salih, “Tripartite counterfactual quantum cryptography,” Phys. Rev. A 90, 012333 (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]

Schuck, C.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997), Sec. 19.3.
[Crossref]

Shin, H.

F. Zaman, Y. Jeong, and H. Shin, “Counterfactual Bell-state analysis,” Scientific Reports 8, 14641 (2018).
[Crossref] [PubMed]

Tang, H. X.

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

Tittel, W.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Urban, E.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Vaidman, L.

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

Walker, T. G.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Wang, H. F.

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

Wang, H.-F.

Weihs, G.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

Weinfurter, H.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

White, A. G.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

Xie, L.

Yavuz, D. D.

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
[Crossref] [PubMed]

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006).
[Crossref] [PubMed]

Yin, H. L.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Yin, J.

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
[Crossref] [PubMed]

Zaman, F.

F. Zaman, Y. Jeong, and H. Shin, “Counterfactual Bell-state analysis,” Scientific Reports 8, 14641 (2018).
[Crossref] [PubMed]

Zeilinger, A.

P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency Quantum Interrogation Measurements via the Quantum Zeno Effect,” Phys. Rev. Lett. 83, 4725–4728 (1999).
[Crossref]

P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995).
[Crossref] [PubMed]

Zhai, S.

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Zhang, J. X.

C. Liu, J. H. Liu, J. X. Zhang, and S.-Y. Zhu, “The experimental demonstration of high efficiency interaction-free measurement for quantum counterfactual-like communication,” Scientific Reports 7, 10875 (2017).
[Crossref] [PubMed]

Zhang, J.-X.

C. Liu, J. -H. Liu, J.-X. Zhang, and S. -Y. Zhu, “Improvement of reliability in multi interferometer-based counterfactual deterministic communication with dissipation compensation,” Opt. Express 26, 2261–2269 (2018).
[Crossref] [PubMed]

F. Li, J.-X. Zhang, and S.-Y. Zhu, “Numerical simulation of the effect of dissipation and phase fluctuation in a direct communication scheme,” J. Phys. B: At. Mol. Opt. Phys. 48, 115506 (2015).
[Crossref]

Zhang, S.

Q. Guo, L.-Y. Cheng, H.-F. Wang, and S. Zhang, “Counterfactual entanglement swapping enables high-efficiency entanglement distribution,” Opt. Express 26, 27314–27325 (2018).
[Crossref] [PubMed]

Q. Guo, S. Zhai, L.-Y. Cheng, H. F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017).
[Crossref]

Q. Guo, L.Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Scientific Reports 5, 8416 (2015).
[Crossref] [PubMed]

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]

Q. Guo, L.-Y. Cheng, L. Chen, H.-F. Wang, and S. Zhang, “Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities,” Phys. Rev. A 90, 042327 (2014).
[Crossref]

Zhu, S. -Y.

Zhu, S.-Y.

C. Liu, J. H. Liu, J. X. Zhang, and S.-Y. Zhu, “The experimental demonstration of high efficiency interaction-free measurement for quantum counterfactual-like communication,” Scientific Reports 7, 10875 (2017).
[Crossref] [PubMed]

F. Li, J.-X. Zhang, and S.-Y. Zhu, “Numerical simulation of the effect of dissipation and phase fluctuation in a direct communication scheme,” J. Phys. B: At. Mol. Opt. Phys. 48, 115506 (2015).
[Crossref]

Zubairy, M. S.

Z.-H. Li, M. S. Zubairy, and M. Al-Amri, “Quantum secure group communication,” Scientific Reports 8, 3899 (2018).
[Crossref] [PubMed]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015).
[Crossref]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct quantum communication with almost invisible photons,” Phys. Rev. A 89, 052334 (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]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Comment on “Past of a quantum particle”,” Phys. Rev. A 88, 046102 (2013).
[Crossref]

M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997), Sec. 19.3.
[Crossref]

Found. Phys. (1)

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

Front. Phys. (1)

H. Salih, “Protocol for counterfactually transporting an unknown qubit,” Front. Phys. 3, 94 (2016).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. B: At. Mol. Opt. Phys. (1)

F. Li, J.-X. Zhang, and S.-Y. Zhu, “Numerical simulation of the effect of dissipation and phase fluctuation in a direct communication scheme,” J. Phys. B: At. Mol. Opt. Phys. 48, 115506 (2015).
[Crossref]

Nat. Phys. (1)

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

Nature (2)

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref] [PubMed]

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]

Opt. Express (4)

Phys. Rev. A (10)

X.-S. Ma, X. Guo, C. Schuck, K. Y. Fong, L. Jiang, and H. X. Tang, “On-chip interaction-free measurements via the quantum Zeno effect,” Phys. Rev. A 90, 042109 (2014).
[Crossref]

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

D. R. M. Arvidsson-Shukur and C. H. W. Barnes, “Quantum counterfactual communication without a weak trace,” Phys. Rev. A 94, 062303 (2016).
[Crossref]

Z.-H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015).
[Crossref]

M. Pavičić, “Nondestructive interaction-free atom-photon controlled-NOT gate,” Phys. Rev. A 75, 032342 (2007).
[Crossref]

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Proc. Natl. Acad. Sci. USA (1)

Y. Cao, Y. H. Li, Z. Cao, J. Yin, Y. A. Chen, H. L. Yin, T. Y. Chen, X. F. Ma, C. Z. Peng, and J. W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920–4924 (2017).
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Figures (6)

Fig. 1
Fig. 1 The proposed scheme for the QCIP. At Alice’s end, QS represents quantum switch, D represents detector, OD represents optical delay, BS represents beam-splitter, MR represents normal mirror, SM represents switchable mirror and SPR represents switchable polarization rotator. In addition, PBS represents polarizing beam splitter while SPBS represents switchable polarizing beam splitter. Both of them only reflect V photons. At Bob’s end, it is a three-level atom. The transition between its levels |e〉 and |g〉 is driven by a driving field while the transition between the levels |u〉 and |g〉 is dispersively coupled to Alice’s photon with detuning Δ. Between Alice and Bob, it is the public transmission channel.
Fig. 2
Fig. 2 A single-side cavity with a there-level atom inside. The model can give a controllable phase to an incident photon. If the atom is in level |1〉, the photon is reflected with a π phase shift while if the atom is in level |2〉, the photon is reflected without the π phase shift.
Fig. 3
Fig. 3 (a) The proposed scheme for the QCSG, where Alice’s photon and Bob’s atom are operated step by step to exchange their quantum states. Only in steps 1,3,5, the photon and the atom have interaction through QCEO and QCPG. (b) A time-bin operation device (TBO).
Fig. 4
Fig. 4 The photon loss in the transmission channel caused by random blockages. We plot (a) effective fidelity, (b) transfer efficiency, and (c) the total probability of the photon loss in the transmission channel (PC) versus the probability of the transmission channel being blocked γ, for different cycle numbers M and N.
Fig. 5
Fig. 5 (a) Effective fidelity and (b) transfer efficiency versus κ, which is the probability of Alice’s photon being absorbed or scattered each time the photon passing the transmission channel, for different cycle numbers M and N.
Fig. 6
Fig. 6 The influence of the imperfection of mirrors. All mirrors used in Fig. 1 are considered as the same. They have the same transmissivity ξ. We plot for (a) effective fidelity and (b) transfer efficiency versus ξ for different cycle numbers M and N.

Equations (21)

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| P 00 cos π 4 M | P 00 + sin π 4 M | 0 P 0 , | 0 P 0 cos π 4 M | 0 P 0 sin π 4 M | P 00 .
| 0 P 0 cos π 2 N | 0 P 0 + sin π 2 N | 00 P , | 00 P cos π 2 N | 00 P sin π 2 N | 0 P 0 .
| 0 P 0 1 st cos π 2 N | 0 P 0 + sin π 2 N | 00 P π cos π 2 N | 0 P 0 sin π 2 N | 00 P 2 nd | 0 P 0 .
cos n π N | 0 P 0 + sin n π N | 00 P .
( C H | H 00 + C V | V 00 ) ( C e | e + C g | g ) .
C H C e [ cos ( 2 m 1 ) π 4 M | H 00 + sin ( 2 m 1 ) π 4 M ( cos π 2 N | 0 H 0 + sin π 2 N | 00 H ) ] | e + C H C g [ cos π 4 M | H 00 + sin π 4 M ( cos ( 2 n 1 ) π 2 N | 0 H 0 + sin ( 2 n 1 ) π 2 N | 00 H ) ] | g + C V | V 00 ( C e | e + C g | g ) .
| C H C e sin ( 2 m 1 ) π 4 M sin π 2 N | 2 + | C H C g sin π 4 M sin ( 2 n 1 ) π 2 N | 2 .
C H C e ( cos m π 2 M | H 00 + sin m π 2 M | 0 H 0 ) | e + C H C g | H 00 | g + C V | V 00 ( C e | e + C g | g ) .
C H C e ( cos m π 2 M | H 00 + sin m π 2 M | 0 V 0 ) | e + C H C g | H 00 | g + C V | V 00 ( C e | e + C g | g ) .
| H ( C e | e + C g | g ) C e | V | e + C g | H | g .
( C H | H + C V | V ) ( C e | e + C g | g ) C H | H ( C g | g C e | e ) + C V | V ( C g | g + C e | e ) .
m = 1 M sin 2 ( m π 2 M ) [ 1 cos 2 N ( π N ) ] [ 1 cos 2 N ( π N ) ] 2 M π 0 π 2 sin 2 m d m = M 2 [ 1 cos 2 N ( π N ) ] M π 2 2 N .
| e ( | e | g ) / 2 , | g ( | e + | g ) / 2 .
| H cos θ | H + sin θ | V , | V cos θ | V sin θ | H ,
f 1 ( C e | e | V 1 + C g | g | H 1 ) + f 2 ( C e | e | V 2 + C g | g | H 2 ) .
C g f 1 2 | H 1 ( | e + | g ) + C e f 1 2 | V 1 ( | e | g ) + C g f 2 2 | V 2 ( | e + | g ) + C e f 2 2 | H 2 ( | e | g ) .
C g f 1 2 | H 1 ( | g | e ) + C e f 1 2 | V 1 ( | e | g ) + C g f 2 2 | V 2 ( | g + | e ) + C e f 2 2 | H 2 ( | e | g ) .
C g f 1 2 ( | H 1 + | V 1 ) | g C e f 1 2 ( | H 2 + | V 2 ) | g + C g f 2 2 ( | H 1 + | V 1 ) | e C e f 2 2 ( | H 2 + | V 2 ) | e .
C g f 1 2 ( | H 1 + | V 1 ) | g + C g f 2 2 ( | H 1 + | V 1 ) | e + C e f 1 2 ( | H 2 | V 2 ) | g + C e f 2 2 ( | H 2 | V 2 ) | e .
( f 1 | g + f 2 | e ) ( C g | H 1 + C e | H 2 ) .
cos γ ( N 1 ) + 1 N γ ( N 1 ) + 1 π N 1 π 2 2 ( γ N + 1 ) .

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