Abstract

The decoy-state high-dimensional quantum key distribution provides a practical secure way to share more private information with high photon-information efficiency. In this paper, based on detector-decoy method, we propose a detector-decoy high-dimensional quantum key distribution protocol. Employing threshold detectors and a variable attenuator, we can promise the security under Gsussian collective attacks with much simpler operations in practical implementation. By numerical evaluation, we show that without varying the source intensity, our protocol performs much better than one-decoy-state protocol and as well as the two-decoy-state protocol in the infinite-size regime. In the finite-size regime, our protocol can achieve better results. Specially, when the detector efficiency is lower, the advantage of the detector-decoy method becomes more prominent.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
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  4. S. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
    [Crossref] [PubMed]
  5. H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
    [Crossref] [PubMed]
  6. S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nature Photon. 9, 397–402 (2015).
    [Crossref]
  7. C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
    [Crossref]
  8. R. K. Chen, W. S. Bao, Y. Wang, H. Z. Bao, C. Zhou, and H. W. Li, “Biased decoy-state measurement-device-independent quantum cryptographic conferencing with finite resources,” Opt. Express 24, 6594 (2016).
    [Crossref] [PubMed]
  9. T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature (London) 509, 475 (2014).
    [Crossref]
  10. L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
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  11. 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).
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  12. R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
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  13. B. Qi, “Single-photon continuous-variable quantum key distribution based on the energy-time uncertainty relation,” Opt. Lett. 31, 2795 (2006).
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  14. I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
    [Crossref] [PubMed]
  15. J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, “Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion,” Opt. Express 21, 15959 (2013).
    [Crossref] [PubMed]
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  17. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
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  22. Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
    [Crossref] [PubMed]
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    [Crossref]
  24. N. Lütkenhaus and M. Jahma, “Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack,” New J. Phys. 4(1), 44 (2002).
    [Crossref]
  25. D. Somma and R. Hughes, “Security of decoy-state protocols for general photon-number-splitting attacks,” Phys. Rev. A 87, 062330 (2013).
    [Crossref]
  26. W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
    [Crossref] [PubMed]
  27. H.-K. Lo, X.-F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2004).
    [Crossref]
  28. X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
    [Crossref]
  29. X.-F. Ma, B. Qi, Y. Zhao, and H.-K. Lo, ”Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005).
    [Crossref]
  30. Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Simple and efficient quantum key distribution with parametric down-conversion,” Phys. Rev. Lett. 99, 180503 (2007).
    [Crossref] [PubMed]
  31. C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
    [Crossref]
  32. Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
    [Crossref]
  33. D. Bunandar, Z. Zhang, J. H. Shapiro, and D. Englund, “Practical high-dimensional quantum key distribution with decoy states,” Phys. Rev. A 91, 022336 (2015).
    [Crossref]
  34. H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
    [Crossref]
  35. T. Moroder, M. Curty, and N. Lütkenhaus, “Detector decoy quantum key distribution,” New J. Phys. 11, 045008 (2009).
    [Crossref]
  36. I. Ali-Khan and J. C. Howell, “Experimental demonstration of high two-photon time-energy entanglement,” Phys. Rev. A 73, 031801 (2006).
    [Crossref]
  37. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
    [Crossref] [PubMed]
  38. D. Deutsch, A.A Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818 (1996).
    [Crossref] [PubMed]
  39. D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325 (2004).
  40. X.-S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83, 043814 (2011).
    [Crossref]
  41. C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
    [Crossref]
  42. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
    [Crossref]
  43. C. Lee, J. Mower, Z. Zhang, J. H. Shapiro, and D. Englund, “Finite-key analysis of high-dimensional time-energy entanglement-based quantum key distribution,” Quantum Inf. Process. 14, 1005 (2015).
    [Crossref]
  44. C. Gobby, Z.-L. Yuan, and A. J. Shields, “Quantum key distribution over 122 km of standard telecom fiber,” Appl. Phys. Lett. 84, 3762 (2004).
    [Crossref]

2016 (2)

R. K. Chen, W. S. Bao, Y. Wang, H. Z. Bao, C. Zhou, and H. W. Li, “Biased decoy-state measurement-device-independent quantum cryptographic conferencing with finite resources,” Opt. Express 24, 6594 (2016).
[Crossref] [PubMed]

H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
[Crossref]

2015 (4)

D. Bunandar, Z. Zhang, J. H. Shapiro, and D. Englund, “Practical high-dimensional quantum key distribution with decoy states,” Phys. Rev. A 91, 022336 (2015).
[Crossref]

C. Lee, J. Mower, Z. Zhang, J. H. Shapiro, and D. Englund, “Finite-key analysis of high-dimensional time-energy entanglement-based quantum key distribution,” Quantum Inf. Process. 14, 1005 (2015).
[Crossref]

S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nature Photon. 9, 397–402 (2015).
[Crossref]

C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
[Crossref]

2014 (6)

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature (London) 509, 475 (2014).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nature Photon. 8(8), 595–604 (2014).
[Crossref]

Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
[Crossref] [PubMed]

C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
[Crossref]

Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
[Crossref]

C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
[Crossref]

2013 (5)

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
[Crossref]

D. Somma and R. Hughes, “Security of decoy-state protocols for general photon-number-splitting attacks,” Phys. Rev. A 87, 062330 (2013).
[Crossref]

J. Mower, Z. Zhang, P. Desjardins, C.C Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, “Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion,” Opt. Express 21, 15959 (2013).
[Crossref] [PubMed]

2012 (2)

S. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
[Crossref] [PubMed]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
[Crossref] [PubMed]

2011 (1)

X.-S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83, 043814 (2011).
[Crossref]

2009 (2)

T. Moroder, M. Curty, and N. Lütkenhaus, “Detector decoy quantum key distribution,” New J. Phys. 11, 045008 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

2008 (1)

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[Crossref] [PubMed]

2007 (2)

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Simple and efficient quantum key distribution with parametric down-conversion,” Phys. Rev. Lett. 99, 180503 (2007).
[Crossref] [PubMed]

2006 (2)

B. Qi, “Single-photon continuous-variable quantum key distribution based on the energy-time uncertainty relation,” Opt. Lett. 31, 2795 (2006).
[Crossref] [PubMed]

I. Ali-Khan and J. C. Howell, “Experimental demonstration of high two-photon time-energy entanglement,” Phys. Rev. A 73, 031801 (2006).
[Crossref]

2005 (2)

X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
[Crossref]

X.-F. Ma, B. Qi, Y. Zhao, and H.-K. Lo, ”Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005).
[Crossref]

2004 (6)

H.-K. Lo, X.-F. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2004).
[Crossref]

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[Crossref] [PubMed]

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325 (2004).

C. Gobby, Z.-L. Yuan, and A. J. Shields, “Quantum key distribution over 122 km of standard telecom fiber,” Appl. Phys. Lett. 84, 3762 (2004).
[Crossref]

2003 (1)

W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
[Crossref] [PubMed]

2002 (3)

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref] [PubMed]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

N. Lütkenhaus and M. Jahma, “Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack,” New J. Phys. 4(1), 44 (2002).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412, 313 (2001).
[Crossref]

2000 (2)

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]

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A 61, 052304 (2000).
[Crossref]

1996 (1)

D. Deutsch, A.A Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818 (1996).
[Crossref] [PubMed]

1984 (1)

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theoretical Comput. Sci. 560, 175–179 (1984).

Acín, A.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Adachi, Y.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Simple and efficient quantum key distribution with parametric down-conversion,” Phys. Rev. Lett. 99, 180503 (2007).
[Crossref] [PubMed]

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

I. Ali-Khan and J. C. Howell, “Experimental demonstration of high two-photon time-energy entanglement,” Phys. Rev. A 73, 031801 (2006).
[Crossref]

Andersen, U. L.

S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nature Photon. 9, 397–402 (2015).
[Crossref]

Baek, B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
[Crossref]

Bao, H. Z.

R. K. Chen, W. S. Bao, Y. Wang, H. Z. Bao, C. Zhou, and H. W. Li, “Biased decoy-state measurement-device-independent quantum cryptographic conferencing with finite resources,” Opt. Express 24, 6594 (2016).
[Crossref] [PubMed]

H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
[Crossref]

Bao, W. S.

H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
[Crossref]

R. K. Chen, W. S. Bao, Y. Wang, H. Z. Bao, C. Zhou, and H. W. Li, “Biased decoy-state measurement-device-independent quantum cryptographic conferencing with finite resources,” Opt. Express 24, 6594 (2016).
[Crossref] [PubMed]

C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
[Crossref]

Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
[Crossref]

C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theoretical Comput. Sci. 560, 175–179 (1984).

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theoretical Comput. Sci. 560, 175–179 (1984).

Braunstein, S.

S. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
[Crossref] [PubMed]

Braunstein, S. L.

S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nature Photon. 9, 397–402 (2015).
[Crossref]

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]

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Bunandar, D.

D. Bunandar, Z. Zhang, J. H. Shapiro, and D. Englund, “Practical high-dimensional quantum key distribution with decoy states,” Phys. Rev. A 91, 022336 (2015).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Chen, K.

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D. Deutsch, A.A Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818 (1996).
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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
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T. Moroder, M. Curty, and N. Lütkenhaus, “Detector decoy quantum key distribution,” New J. Phys. 11, 045008 (2009).
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C. Lee, J. Mower, Z. Zhang, J. H. Shapiro, and D. Englund, “Finite-key analysis of high-dimensional time-energy entanglement-based quantum key distribution,” Quantum Inf. Process. 14, 1005 (2015).
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Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
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C. Lee, J. Mower, Z. Zhang, J. H. Shapiro, and D. Englund, “Finite-key analysis of high-dimensional time-energy entanglement-based quantum key distribution,” Quantum Inf. Process. 14, 1005 (2015).
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Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
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C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
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R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412, 313 (2001).
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C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
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Wang, L.

C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
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Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
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C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
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Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
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C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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C. Gobby, Z.-L. Yuan, and A. J. Shields, “Quantum key distribution over 122 km of standard telecom fiber,” Appl. Phys. Lett. 84, 3762 (2004).
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R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
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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).
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X.-S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83, 043814 (2011).
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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412, 313 (2001).
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Zhang, H. L.

C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
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Zhang, L.

Zhang, Z.

D. Bunandar, Z. Zhang, J. H. Shapiro, and D. Englund, “Practical high-dimensional quantum key distribution with decoy states,” Phys. Rev. A 91, 022336 (2015).
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C. Lee, J. Mower, Z. Zhang, J. H. Shapiro, and D. Englund, “Finite-key analysis of high-dimensional time-energy entanglement-based quantum key distribution,” Quantum Inf. Process. 14, 1005 (2015).
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C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
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J. Mower, Z. Zhang, P. Desjardins, C.C Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
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Zhao, Y.

X.-F. Ma, B. Qi, Y. Zhao, and H.-K. Lo, ”Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005).
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Zhong, T.

C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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Zhou, C.

H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
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R. K. Chen, W. S. Bao, Y. Wang, H. Z. Bao, C. Zhou, and H. W. Li, “Biased decoy-state measurement-device-independent quantum cryptographic conferencing with finite resources,” Opt. Express 24, 6594 (2016).
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C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
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Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
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C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
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Zhou, H.

C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
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Zotter, S.

X.-S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83, 043814 (2011).
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Appl. Phys. Lett. (1)

C. Gobby, Z.-L. Yuan, and A. J. Shields, “Quantum key distribution over 122 km of standard telecom fiber,” Appl. Phys. Lett. 84, 3762 (2004).
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J. Phys. A: Math. Theor. (1)

H. Z. Bao, W. S. Bao, Y. Wang, C. Zhou, and R. K. Chen, ”Finite-key analysis of a practical decoystate high-dimensional quantum key distribution,” J. Phys. A: Math. Theor. 49, 205301 (2016).
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Nature (London) (2)

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature (London) 509, 475 (2014).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412, 313 (2001).
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Nature Photon. (3)

S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “High-rate measurement-device-independent quantum cryptography,” Nature Photon. 9, 397–402 (2015).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nature Photon. 8(8), 595–604 (2014).
[Crossref]

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photon. 7, 210 (2013).
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New J. Phys. (2)

T. Moroder, M. Curty, and N. Lütkenhaus, “Detector decoy quantum key distribution,” New J. Phys. 11, 045008 (2009).
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N. Lütkenhaus and M. Jahma, “Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack,” New J. Phys. 4(1), 44 (2002).
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Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (13)

D. Somma and R. Hughes, “Security of decoy-state protocols for general photon-number-splitting attacks,” Phys. Rev. A 87, 062330 (2013).
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N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A 61, 052304 (2000).
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X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
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X.-F. Ma, B. Qi, Y. Zhao, and H.-K. Lo, ”Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005).
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I. Ali-Khan and J. C. Howell, “Experimental demonstration of high two-photon time-energy entanglement,” Phys. Rev. A 73, 031801 (2006).
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C. Zhou, W. S. Bao, H. W. Li, Y. Wang, Y. Li, Z. Q. Yin, W. Chen, and Z. F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89(5), 052328 (2014).
[Crossref]

Y. Li, W. S. Bao, H. W. Li, C. Zhou, and Y. Wang, “Passive decoy-state quantum key distribution using weak coherent pulses with intensity fluctuations,” Phys. Rev. A 89, 032329 (2014).
[Crossref]

D. Bunandar, Z. Zhang, J. H. Shapiro, and D. Englund, “Practical high-dimensional quantum key distribution with decoy states,” Phys. Rev. A 91, 022336 (2015).
[Crossref]

X.-S. Ma, S. Zotter, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental generation of single photons via active multiplexing,” Phys. Rev. A 83, 043814 (2011).
[Crossref]

C. Lee, Z. Zhang, R. Steinbrecher, H. Zhou, J. Mower, T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E. Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wornell, F. N. C. Wong, J. H. Shapiro, and D. Englund, “Entanglement-based quantum communication secured by nonlocal dispersion cancellation,” Phys. Rev. A 90, 062331 (2014).
[Crossref]

C. Zhou, W. S. Bao, H. L. Zhang, H. W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015).
[Crossref]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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J. Mower, Z. Zhang, P. Desjardins, C.C Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

Phys. Rev. Lett. (15)

Z. Zhang, J.J Mower, D. Englund, F. Wong, and J. H. Shapiro, “Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry,” Phys. Rev. Lett. 112, 120506 (2014).
[Crossref] [PubMed]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[Crossref] [PubMed]

S. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
[Crossref] [PubMed]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
[Crossref] [PubMed]

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008).
[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]

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Schematic of the detector-decoy HD-QKD setup. The source is located in Alice’s side. Alice keeps one photon of the pair and sends the other to Bob. They both choose to measure in either the arrival-time basis (case 1) or the frequency basis (case 2) independently and randomly. Their results are only correlated or anti-correlated when they choose the same basis. VA is a variable attenuator. ND is normal dispersion and AD is anomalous dispersion.
Fig. 2
Fig. 2 Lower bounds on the secure-key capacity in bits per coincidence for (a) d = 8 and (b) d = 32 as a function of transmission distance at 10 km increments in the infinte-size regime. The results show that the detector-decoy method (black dash lines) could perform much better than one-decoy-state HD-QKD (blue dash lines) and as well as the two-decoy-state HD-QKD (red dotted lines).
Fig. 3
Fig. 3 Lower bounds on the secure-key capacity in the finite-size regime for pulse number N = 1012 and d = 8. The results show that the detector-decoy method (black dash lines) could perform much better than two-decoy-state HD-QKD (red dash lines).
Fig. 4
Fig. 4 (a) The lower bound values on Fμ obtained by detector-decoy method (blue solid lines) and two-decoy-state HD-QKD (red solid lines). (b) The secret key capacity per coincidence for original decoy-state HD-QKD (red dotted lines) and detector-decoy HD-QKD (black solid lines) as a function of the transmission distance when d = 8, μ = 0.10 and ηAlice = ηBob = 0.045.

Equations (24)

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P μ = n = 0 Pr n γ n .
γ n = α n ( η ) β n = [ 1 ( 1 η η Alice ) n ( 1 p d ) ] [ 1 ( 1 η Bob η T ) n ( 1 p d ) ] .
Δ I β I ( A ; B ) ( 1 F μ ( η ) ) n R F μ ( η ) χ ζ t , ζ ω U B ( A ; E ) .
Pr n = μ n n ! e μ .
P μ ( η 1 ) = n = 0 Pr n α n ( η 1 ) β n = n = 0 μ n n ! e μ ( 1 ( 1 η 1 η Alice ) n ( 1 p d ) ) β n ,
P μ ( η 2 ) = n = 0 Pr n α n ( η 2 ) β n = n = 0 μ n n ! e μ ( 1 ( 1 η 2 η Alice ) n ( 1 p d ) ) β n .
e μ P μ ( η 1 ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) e μ P μ ( η 2 ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) = β 0 [ p d 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) p d 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) ] + β 1 [ 1 ( 1 η 1 η Alice ) ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) μ 1 ( 1 η 2 η Alice ) ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) μ ] + n = 3 β n [ 1 ( 1 η 1 η Alice ) n ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) μ n 1 ( 1 η 2 η Alice ) n ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) μ n ] .
1 ( 1 η 1 η Alice ) n ( 1 p d ) 1 ( 1 η 2 η Alice ) n ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d )
n = 3 β n [ 1 ( 1 η 1 η Alice ) n ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) μ n 1 ( 1 η 2 η Alice ) n ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) μ n ] 0 .
β 1 β 1 L B = e μ P μ ( η 1 ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) e μ P μ ( η 2 ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) + β 0 [ p d 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) p d 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) ] 1 ( 1 η 1 η Alice ) ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) 1 ( 1 η 2 η Alice ) ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) .
β 0 β 0 L B = p d .
F μ ( η 1 ) = α 1 ( η 1 ) β 1 μ e μ P μ ( η 1 ) F μ L B = ( 1 ( 1 η 1 η Alice ) ( 1 p d ) ) × P μ ( η 1 ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) P μ ( η 2 ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) + e μ β 0 L B [ p d 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) p d 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) ] P μ ( η 1 ) [ 1 ( 1 η 1 η Alice ) ( 1 p d ) 1 ( 1 η 1 η Alice ) 2 ( 1 p d ) 1 ( 1 η 2 η Alice ) ( 1 p d ) 1 ( 1 η 2 η Alice ) 2 ( 1 p d ) ]
Ω x = F μ ( η ) ( 1 + ζ x ) + Δ Ω x ( 1 F μ ( η ) ) ,
Ω x , η 1 = F μ ( η 1 ) ( 1 + ζ x ) + Δ Ω x ( 1 F μ ( η 1 ) ) ,
Ω x , η 2 = F μ ( η 2 ) ( 1 + ζ x ) + Δ Ω x ( 1 F μ ( η 2 ) ) .
Ω x , η 1 P μ ( η 1 ) e μ = μ α 1 ( η 1 ) β 1 ( 1 + ζ x ) + Δ Ω x ( P μ ( η 1 ) e μ μ α 1 ( η 1 ) β 1 ) ,
Ω x , η 2 P μ ( η 2 ) e μ = μ α 1 ( η 2 ) β 1 ( 1 + ζ x ) + Δ Ω x ( P μ ( η 2 ) e μ μ α 1 ( η 2 ) β 1 ) .
Ω x , η 1 P μ ( η 1 ) e μ Ω x , η 2 P μ ( η 2 ) e μ = ( α 1 ( η 1 ) α 1 ( η 2 ) ) μ β 1 ( 1 + ζ x ) + Δ Ω x [ P μ ( η 1 ) e u P μ ( η 2 ) e u ( α 1 ( η 1 ) α 1 ( η 2 ) ) μ β 1 ] = ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 ( 1 + ζ x ) + Δ Ω x [ P μ ( η 1 ) e u P μ ( η 2 ) e u ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 ] ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 ( 1 + ζ x ) ,
P μ ( η 1 ) P μ ( η 2 ) = n = 0 { [ ( 1 η 2 η Alice ) n ( 1 η 1 η Alice ) n ] ( 1 p d ) } β n ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 .
( 1 + ζ x ) ( Ω x , η 1 P μ ( η 1 ) Ω x , η 2 P μ ( η 2 ) ) e μ ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 ( Ω x , η 1 P μ ( η 1 ) Ω x , η 2 P μ ( η 2 ) ) e μ ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 L B
Ω x , η = F μ ( η ) ( 1 + ζ x ) + Δ Ω x ( 1 F μ ( η ) ) F μ ( η ) ( 1 + ζ x ) = [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η 1 ) [ 1 ( 1 η 1 η Alice ) ( 1 p d ) ] P μ ( η ) F μ ( η 1 ) ( 1 + ζ x ) [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η 1 ) [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η ) F μ L B ( 1 + ζ x )
( 1 + ζ x ) [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η ) Ω x , η [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η 1 ) F μ L B .
ζ x ζ x U B = min { ( Ω x , η 1 P μ ( η 1 ) Ω x , η 2 P μ ( η 2 ) ) e μ ( η 1 η 2 ) η Alice ( 1 p d ) μ β 1 L B , min η { η 1 , η 2 } { [ 1 ( 1 η 1 η Alice ) ( 1 p d ) ] P μ ( η ) Ω x , η [ 1 ( 1 η η Alice ) ( 1 p d ) ] P μ ( η 1 ) F μ L B } } 1 .
P μ ( η ) ± = P μ ( η ) ± Δ .

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