Abstract

Recently Zhang et al [ Phys. Rev. A 95, 012333 (2017)] developed a new approach to estimate the failure probability for the decoy-state BB84 QKD system when taking finite-size key effect into account, which offers security comparable to Chernoff bound, while results in an improved key rate and transmission distance. Based on Zhang et al’s work, now we extend this approach to the case of the measurement-device-independent quantum key distribution (MDI-QKD), and for the first time implement it onto the four-intensity decoy-state MDI-QKD system. Moreover, through utilizing joint constraints and collective error-estimation techniques, we can obviously increase the performance of practical MDI-QKD systems compared with either three- or four-intensity decoy-state MDI-QKD using Chernoff bound analysis, and achieve much higher level security compared with those applying Gaussian approximation analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  11. B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).
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  16. I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
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  23. M. Curty and T. Moroder, “Heralded-qubit amplifiers for practical device-independent quantum key distribution,” Phys. Rev. A. 84, 010304 (2011).
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  24. S. L. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
    [Crossref] [PubMed]
  25. H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
    [Crossref] [PubMed]
  26. X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013).
    [Crossref]
  27. Q. Wang and X.-B. Wang, “Efficient implementation of the decoy-state measurement-deviceindependent quantum key distribution with heralded single-photon sources,” Phys. Rev. A 88, 052332 (2013).
    [Crossref]
  28. Q. Wang and X.-B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4, 04612 (2014).
    [Crossref]
  29. H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
    [Crossref]
  30. C. Wang, X. T. Song, Z. Q. Yin, S. Wang, W. Chen, C. M. Zhang, G. C. Guo, and Z. F. Han, “Phase-reference-free experiment of measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 115, 160502 (2015).
    [Crossref] [PubMed]
  31. C. Wang, S. Wang, Z. Q. Yin, W. Chen, H. W. Li, C. M. Zhang, Y. Y. Ding, G. C. Guo, and Z. F. Han, “Experimental measurement-device-independent quantum key distribution with uncharacterized encoding,” Opt. Lett. 41, 5596–5599 (2016).
    [Crossref] [PubMed]
  32. C. Wang, Z. Q. Yin, S. Wang, W. Chen, G. C. Guo, and Z. F. Han, “Measurement-device-independent quantum key distribution robust against environmental disturbances,” Optica 4, 1016–1023 (2017).
    [Crossref]
  33. Y. H. Zhou, Z. W. Yu, and X.-B. Wang, “Making the decoy-state measurement-deviceindependent quantum key distribution practically useful,” Phys. Rev. A,  93, 042324 (2016).
    [Crossref]
  34. X. Y. Zhou, C. H. Zhang, C. M. Zhang, and Q. Wang, “Obtaining better performance in the measurement-device-independent quantum key distribution with heralded single-photon sources,” Phys. Rev. A,  96, 052337 (2017).
    [Crossref]
  35. X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
    [Crossref]
  36. C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
    [Crossref]
  37. Y. Wang, W.-S. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, “Tight finite-key analysis of a practical decoy-state quantum key distribution with unstable sources,” Phys. Rev. A 94, 032335 (2016).
    [Crossref]
  38. M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
    [Crossref] [PubMed]
  39. H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
    [Crossref]
  40. Z. Zhang, Q. Zhao, M. Razavi, and X. Ma, “Improved key-rate bounds for practical decoy-state quantum-key-distribution systems,” Phys. Rev. A 95, 012333 (2017).
    [Crossref]
  41. D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325–360 (2004).
  42. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
    [Crossref]
  43. F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
    [Crossref]

2017 (3)

C. Wang, Z. Q. Yin, S. Wang, W. Chen, G. C. Guo, and Z. F. Han, “Measurement-device-independent quantum key distribution robust against environmental disturbances,” Optica 4, 1016–1023 (2017).
[Crossref]

X. Y. Zhou, C. H. Zhang, C. M. Zhang, and Q. Wang, “Obtaining better performance in the measurement-device-independent quantum key distribution with heralded single-photon sources,” Phys. Rev. A,  96, 052337 (2017).
[Crossref]

Z. Zhang, Q. Zhao, M. Razavi, and X. Ma, “Improved key-rate bounds for practical decoy-state quantum-key-distribution systems,” Phys. Rev. A 95, 012333 (2017).
[Crossref]

2016 (4)

Y. Wang, W.-S. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, “Tight finite-key analysis of a practical decoy-state quantum key distribution with unstable sources,” Phys. Rev. A 94, 032335 (2016).
[Crossref]

C. Wang, S. Wang, Z. Q. Yin, W. Chen, H. W. Li, C. M. Zhang, Y. Y. Ding, G. C. Guo, and Z. F. Han, “Experimental measurement-device-independent quantum key distribution with uncharacterized encoding,” Opt. Lett. 41, 5596–5599 (2016).
[Crossref] [PubMed]

Y. H. Zhou, Z. W. Yu, and X.-B. Wang, “Making the decoy-state measurement-deviceindependent quantum key distribution practically useful,” Phys. Rev. A,  93, 042324 (2016).
[Crossref]

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

2015 (2)

C. Wang, X. T. Song, Z. Q. Yin, S. Wang, W. Chen, C. M. Zhang, G. C. Guo, and Z. F. Han, “Phase-reference-free experiment of measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 115, 160502 (2015).
[Crossref] [PubMed]

H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
[Crossref]

2014 (4)

F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
[Crossref]

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
[Crossref] [PubMed]

C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
[Crossref]

Q. Wang and X.-B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4, 04612 (2014).
[Crossref]

2013 (2)

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013).
[Crossref]

Q. Wang and X.-B. Wang, “Efficient implementation of the decoy-state measurement-deviceindependent quantum key distribution with heralded single-photon sources,” Phys. Rev. A 88, 052332 (2013).
[Crossref]

2012 (3)

S. L. 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]

X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
[Crossref]

2011 (3)

M. Curty and T. Moroder, “Heralded-qubit amplifiers for practical device-independent quantum key distribution,” Phys. Rev. A. 84, 010304 (2011).
[Crossref]

N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
[Crossref] [PubMed]

I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
[Crossref] [PubMed]

2010 (2)

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
[Crossref]

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

2008 (2)

Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008).
[Crossref]

V. Makarov and J. Skaar, “Faked states attack using detector efficiency mismatch on SARG04, phase-time, DPSK, and Ekert protocols,” Quantum Inf. Comput. 8, 622–635 (2008).

2007 (2)

B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref] [PubMed]

2006 (1)

V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74, 022313 (2006).
[Crossref]

2005 (3)

X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005).
[Crossref] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

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

2004 (1)

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325–360 (2004).

2003 (1)

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

2002 (1)

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

2001 (1)

D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48, 351–406 (2001).
[Crossref]

2000 (3)

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441 (2000).
[Crossref] [PubMed]

G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
[Crossref] [PubMed]

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

1996 (1)

H. P. Yuen, “Quantum amplifiers, quantum duplicators and quantum cryptography,” Quantum Semiclassical. Opt. 8, 939–949 (1996).
[Crossref]

1995 (1)

B. Huttner, N. Imoto, N. Gisin, and T. Mor, “Quantum cryptography with coherent states,” Phys. Rev. A 51, 1863 (1995).
[Crossref] [PubMed]

1991 (1)

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

Acín, A.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref] [PubMed]

Anisimov, A.

V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74, 022313 (2006).
[Crossref]

Bao, W.-S.

Y. Wang, W.-S. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, “Tight finite-key analysis of a practical decoy-state quantum key distribution with unstable sources,” Phys. Rev. A 94, 032335 (2016).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

Boyd, S. P.

S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Brassard, G.

G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
[Crossref] [PubMed]

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

Braunstein, S. L.

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

Brunner, N.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref] [PubMed]

Chen, C.

Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008).
[Crossref]

Chen, H.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref] [PubMed]

Chen, S. J.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, T. Y.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, W.

Cui, W.

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
[Crossref] [PubMed]

Curty, M.

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
[Crossref] [PubMed]

C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
[Crossref]

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

M. Curty and T. Moroder, “Heralded-qubit amplifiers for practical device-independent quantum key distribution,” Phys. Rev. A. 84, 010304 (2011).
[Crossref]

Ding, Y. Y.

Ekert, A. K.

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

Elser, D.

N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
[Crossref] [PubMed]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
[Crossref]

Fung, C.-H. F.

X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
[Crossref]

Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008).
[Crossref]

B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).

Gao, M.

H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
[Crossref]

Gerhardt, I.

I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
[Crossref] [PubMed]

Gisin, N.

N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
[Crossref] [PubMed]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref] [PubMed]

B. Huttner, N. Imoto, N. Gisin, and T. Mor, “Quantum cryptography with coherent states,” Phys. Rev. A 51, 1863 (1995).
[Crossref] [PubMed]

Gottesman, D.

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Han, Z. F.

Huang, M. Q.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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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, 44 (2002).
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N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
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H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
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Jiang, M.-S.

Y. Wang, W.-S. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, “Tight finite-key analysis of a practical decoy-state quantum key distribution with unstable sources,” Phys. Rev. A 94, 032335 (2016).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
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H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
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Li, H.-W.

Y. Wang, W.-S. Bao, C. Zhou, M.-S. Jiang, and H.-W. Li, “Tight finite-key analysis of a practical decoy-state quantum key distribution with unstable sources,” Phys. Rev. A 94, 032335 (2016).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

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C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
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M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
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M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
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F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
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H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
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Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008).
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B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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X. 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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Lütkenhaus, N.

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325–360 (2004).

N. Lütkenhaus and M. Jahma, “Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack,” New J. Phys. 4, 44 (2002).
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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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G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
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Lydersen, L.

N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
[Crossref] [PubMed]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
[Crossref]

Ma, C.

H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
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Ma, X.

Z. Zhang, Q. Zhao, M. Razavi, and X. Ma, “Improved key-rate bounds for practical decoy-state quantum-key-distribution systems,” Phys. Rev. A 95, 012333 (2017).
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X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
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B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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X. 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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Ma, Z.

H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
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Makarov, V.

I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
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N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
[Crossref] [PubMed]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
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V. Makarov and J. Skaar, “Faked states attack using detector efficiency mismatch on SARG04, phase-time, DPSK, and Ekert protocols,” Quantum Inf. Comput. 8, 622–635 (2008).

V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74, 022313 (2006).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
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G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
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B. Huttner, N. Imoto, N. Gisin, and T. Mor, “Quantum cryptography with coherent states,” Phys. Rev. A 51, 1863 (1995).
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M. Curty and T. Moroder, “Heralded-qubit amplifiers for practical device-independent quantum key distribution,” Phys. Rev. A. 84, 010304 (2011).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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S. L. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
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N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
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D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Inf. Comput. 4, 325–360 (2004).

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441 (2000).
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H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
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Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008).
[Crossref]

B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73–82 (2007).

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

Razavi, M.

Z. Zhang, Q. Zhao, M. Razavi, and X. Ma, “Improved key-rate bounds for practical decoy-state quantum-key-distribution systems,” Phys. Rev. A 95, 012333 (2017).
[Crossref]

X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
[Crossref]

Sanders, B. C.

G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
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N. Gisin, S. Pironio, and N. Sangouard, “Proposal for implementing device-independent quantum key distribution based on a heralded qubit amplifier,” Phys. Rev. Lett. 105, 070501 (2010).
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A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
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P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441 (2000).
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I. Gerhardt, Q. Liu, A. Lamaslinares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011).
[Crossref] [PubMed]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
[Crossref]

V. Makarov and J. Skaar, “Faked states attack using detector efficiency mismatch on SARG04, phase-time, DPSK, and Ekert protocols,” Quantum Inf. Comput. 8, 622–635 (2008).

V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74, 022313 (2006).
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C. Wang, X. T. Song, Z. Q. Yin, S. Wang, W. Chen, C. M. Zhang, G. C. Guo, and Z. F. Han, “Phase-reference-free experiment of measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 115, 160502 (2015).
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M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
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S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
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C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
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Wang, Q.

X. Y. Zhou, C. H. Zhang, C. M. Zhang, and Q. Wang, “Obtaining better performance in the measurement-device-independent quantum key distribution with heralded single-photon sources,” Phys. Rev. A,  96, 052337 (2017).
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Wang, W.

H. Li, H. Jiang, M. Gao, Z. Ma, C. Ma, and W. Wang, “Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions,” Phys. Rev. A 92, 062344 (2015).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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Y. H. Zhou, Z. W. Yu, and X.-B. Wang, “Making the decoy-state measurement-deviceindependent quantum key distribution practically useful,” Phys. Rev. A,  93, 042324 (2016).
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Q. Wang and X.-B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4, 04612 (2014).
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Q. Wang and X.-B. Wang, “Efficient implementation of the decoy-state measurement-deviceindependent quantum key distribution with heralded single-photon sources,” Phys. Rev. A 88, 052332 (2013).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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Wiechers, C.

N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
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L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
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Wittmann, C.

N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” Phys. Rev. Lett. 107, 110501 (2011).
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L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics. 4, 686–689 (2010).
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Xu, F.

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
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C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
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F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
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Xu, H.

F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
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H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
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You, L. X.

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Y. H. Zhou, Z. W. Yu, and X.-B. Wang, “Making the decoy-state measurement-deviceindependent quantum key distribution practically useful,” Phys. Rev. A,  93, 042324 (2016).
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C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014).
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Zhang, C. H.

X. Y. Zhou, C. H. Zhang, C. M. Zhang, and Q. Wang, “Obtaining better performance in the measurement-device-independent quantum key distribution with heralded single-photon sources,” Phys. Rev. A,  96, 052337 (2017).
[Crossref]

Zhang, C. M.

X. Y. Zhou, C. H. Zhang, C. M. Zhang, and Q. Wang, “Obtaining better performance in the measurement-device-independent quantum key distribution with heralded single-photon sources,” Phys. Rev. A,  96, 052337 (2017).
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Figures (4)

Fig. 1
Fig. 1 Comparison for the key generation rates of the four-intensity scheme with different statistical fluctuation methods, in the case of N = 1011. The dashed curve and the dotted curve represent the results of using the Gaussian approximation analysis method and the Chernoff bound method, respectively. The solid curve corresponds to applying the improved statistical fluctuation analysis method.
Fig. 2
Fig. 2 Comparison for the key generation rates of the four-intensity scheme with different statistical fluctuation methods, in the case of N = 1010. The dashed curve and the dotted curve represent the results of using the Gaussian approximation analysis method and the Chernoff bound method, respectively. The solid curve corresponds to implementing the improved statistical fluctuation analysis method.
Fig. 3
Fig. 3 Comparison for the key generation rate between the three-intensity scheme and the four-intensity scheme, both applying the improved new statistical fluctuation analysis method, for N = 1011. The solid curve represents the four-intensity scheme and the dashed curve corresponds to the three-intensity scheme.
Fig. 4
Fig. 4 The optimized key generation rates versus the total number of pulse pairs N for either the three-intensity scheme or four-intensity scheme.

Tables (1)

Tables Icon

Table 1 List of practical parameters for simulations. ηd and Y 0 are the detection efficiency and dark count rate of all detectors, respectively; ed is the alignment error; e 0 corresponds to the error rate of a vacuum pulse; fe denotes the efficiency of error correction; α is the transmission fiber loss constant; ε represents the failure probability.

Equations (36)

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Q lr = n , m = 0 P n l P m r Y nm ,
E lr Q lr = n , m = 0 P n l P m r e nm Y nm ,
𝒫 n l = N l e l l n / n ! λ { o , w , v , μ } N λ e λ λ n / n ! , = p l e l l n / n ! λ { o , w , v , μ } p λ e λ λ n / n ! .
M nm lr 𝒫 n l 𝒫 m r M nm ,
e nm lr M nm lr 𝒫 n l 𝒫 m r e nm M nm ,
M lr n , m = 0 𝒫 n l 𝒫 m r M nm ,
E lr M lr n , m = 0 𝒫 n l 𝒫 m r e nm M nm .
𝔼 [ M nm lr ] = 𝒫 n l 𝒫 m r M nm ,
𝔼 [ e nm lr M nm lr ] = 𝒫 n l 𝒫 m r e nm M nm ,
𝔼 [ M lr ] = n , m = 0 𝒫 n l 𝒫 m r M nm ,
𝔼 [ E lr M lr ] = n , m = 0 𝒫 n l 𝒫 m r e nm M nm ,
𝔼 [ Q lr ] = 𝔼 [ M lr N lr ] = 𝔼 [ M lr ] N lr = n , m = 0 𝒫 n l 𝒫 m r M nm N lr = n , m = 0 p l e l l n / n ! ( p μ e μ μ n + p v e v v n + p w e w w n ) / n ! p r e r r m / m ! ( p μ e μ μ m + p v e v v m + p w e w w m ) / m ! M nm N p l p r = n , m = 0 e ( l + r ) l n r m n ! m ! M nm ( p μ e μ μ n + p v e v v n + p w e w w n ) / n ! ( p μ e μ μ m + p v e v v m + p w e w w m ) / m ! N = n , m = 0 e ( l + r ) l n r m n ! m ! Y nm * ,
𝔼 [ T lr ] = n , m = 0 e ( l + r ) l n r m n ! m ! e nm Y nm * ,
Y nm * = M nm ( p μ e μ μ n + p v e v v n + p w e w w n ) / n ! ( p μ e μ μ m + p v e v v m + p w e w w m ) / m ! N ,
Y 11 * Y 11 * L ( ) = { P 1 v P 2 v 𝔼 [ Q w w ] + P 1 w P 2 w P 0 v 𝔼 [ Q o v ] + P 1 w P 2 w P 0 v 𝔼 [ Q v o ] } P 1 w P 1 v ( P 1 w P 2 v P 1 v P 2 w ) { P 1 w P 2 w 𝔼 [ Q v v ] + P 1 w P 2 w ( P 0 v ) 2 𝔼 [ Q o o ] } P 1 w P 1 v ( P 1 w P 2 v P 1 v P 2 w ) P 1 v P 2 v P 1 w P 1 v ( P 1 w P 2 v P 1 v P 2 w )
M 11 L ( ) = Y 11 * L ( ) ( p μ e μ μ + p v e v v + p w e w w ) ( p μ e μ μ + p v e v v + p w e w w ) N ,
e 11 ph e 11 ph , U ( ) = 𝔼 [ T w w ] / 2 ( P 1 w ) 2 Y 11 * L ,
𝔼 L [ X ] = 0 ,
𝔼 U [ X ] = b ,
𝔼 L [ X ] = X 1 + δ L ( X ) ,
𝔼 U [ X ] = X 1 δ U ( X ) ,
( e δ L ( 1 + δ L ) 1 + δ L ) X 1 + δ L = ε 2 ,
( e δ U ( 1 δ U ) 1 δ U ) X 1 δ U = ε 2 .
δ L = δ U = 3 b + 8 b X + b 2 2 ( X b ) .
N w w 𝔼 [ Q w w ] + N o v 𝔼 [ Q o v ] ( N w w Q w w + N o v Q o v ) / ( 1 + δ w w , o v L ) , N w w 𝔼 [ Q w w ] + N v o 𝔼 [ Q v o ] ( N w w Q w w + N v o Q v o ) / ( 1 + δ w w , v o L ) , N v o 𝔼 [ Q v o ] + N o v 𝔼 [ Q o v ] ( N v o Q v o + N o v Q o v ) / ( 1 + δ v o , o v L ) , N v v 𝔼 [ Q v v ] + N o o 𝔼 [ Q o o ] ( N v v Q v v + N o o Q o o ) / ( 1 δ v v , o o U ) , N w w 𝔼 [ Q w w ] + N v o 𝔼 [ Q v o ] + N o v 𝔼 [ Q o v ] N w w Q w w + N v o Q v o + N o v Q o v 1 + δ w w , v o , o v L , N ow Q ow + N w o Q w o 1 δ o w , w o U N w o 𝔼 [ Q w o ] + N ow 𝔼 [ Q ow ] N ow Q ow + N w o Q w o 1 + δ o w , w o L .
P 1 v P 2 v 𝔼 [ Q w w ] + P 1 w P 2 w P 0 v 𝔼 [ Q o v ] + P 1 w P 2 w P 0 v 𝔼 [ Q v o ] { P 1 v P 2 v N w w 𝒢 + ( P 1 w P 2 w P 0 v N o v P 1 v P 2 v N w w ) N o v Q o v + N v o Q v o 1 + δ o v , v o L , P 1 w P 2 w P 0 v N o v P 1 v P 2 v N w w ; P 1 w P 2 w P 0 v N o v 𝒢 + ( P 1 v P 2 v N w w P 1 w P 2 w P 0 v N o v ) N w w Q w w 1 + δ w w L , P 1 w P 2 w P 0 v N o v < P 1 v P 2 v N w w ,
P 1 w P 2 w 𝔼 [ Q v v ] + P 1 w P 2 w ( P 0 v ) 2 𝔼 [ Q o o ] { P 1 w P 2 w N v v 𝒦 + ( P 1 w P 2 w P 0 v P 0 v N o o P 1 w P 2 w N v v ) N o o Q o o 1 δ o o U , P 1 w P 2 w P 0 v P 0 v N o o P 1 w P 2 w N v v ; P 1 w P 2 w P 0 v P 0 v N o o 𝒦 + ( P 1 w P 2 w N v v P 1 w P 2 w P 0 v P 0 v N o o ) N v v Q v v 1 δ v v U , P 1 w P 2 w P 0 v P 0 v N o o < P 1 w P 2 w N v v ,
e 11 ph e 11 ph , U ( ) = T w w / ( 1 δ w w U ) / 2 ( P 1 w ) 2 Y 11 * L .
Pr ( | X X ¯ | δ X ¯ ) 2 e δ 2 X ¯ / ( 2 + δ ) ,
X L = ( 1 δ ) X ¯ ,
X U = ( 1 + δ ) X ¯ ,
δ = ln ( ε / 2 ) + [ ln ( ε / 2 ) ] 2 8 ln ( ε / 2 ) X ¯ 2 X ¯ ,
R ( ) = M 11 μ μ , L ( ) [ 1 H ( e 11 ph , U ( ) ) ] f e M μ μ H ( E μ μ ) N ,
R = min I R ( ) , ( I = [ L , U ] ) ,
L = P 0 w N ow N ow Q ow + N w o Q w o 1 + δ o w , w o L ( P 0 w ) 2 N o o N o o Q o o 1 δ o o U ,
U = P 0 w N ow N ow Q ow + N w o Q w o 1 δ o w , w o U ( P 0 w ) 2 N o o N o o Q o o 1 + δ o o L .

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