Abstract

We present a detailed description of a widely applicable mathematical model for quantum key distribution (QKD) systems implementing the measurement-device-independent (MDI) protocol. The model is tested by comparing its predictions with data taken using a proof-of-principle, time-bin qubit-based QKD system in a secure laboratory environment (i.e. in a setting in which eavesdropping can be excluded). The good agreement between the predictions and the experimental data allows the model to be used to optimize mean photon numbers per attenuated laser pulse, which are used to encode quantum bits. This in turn allows optimization of secret key rates of existing MDI-QKD systems, identification of rate-limiting components, and projection of future performance. In addition, we also performed measurements over deployed fiber, showing that our system’s performance is not affected by environment-induced perturbations.

© 2014 Optical Society of America

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  1. N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [CrossRef]
  2. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, M. Peev, “The security of practical quantum key disitrbution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
    [CrossRef]
  3. A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
    [CrossRef]
  4. D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
    [CrossRef]
  5. T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
    [CrossRef] [PubMed]
  6. L. Masanes, S. Pironio, A. Acín, “Secure device-independent quantum key distribution with causally independent measurement devices,” Nat. Commun. 2, 238 (2011).
    [CrossRef] [PubMed]
  7. H.-K. Lo, M. Curty, B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
    [CrossRef] [PubMed]
  8. S. L. Braunstein, S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
    [CrossRef] [PubMed]
  9. A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
    [CrossRef] [PubMed]
  10. Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
    [CrossRef] [PubMed]
  11. T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
    [CrossRef]
  12. Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].
  13. G. Brassard, N. Lütkenhaus, T. Mor, B. Sanders, “Limitation on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330 (2000).
    [CrossRef] [PubMed]
  14. W. Hwang, “Quantum key distribution with high loss: towards global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
    [CrossRef]
  15. H.-K. Lo, X. Ma, K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
    [CrossRef] [PubMed]
  16. X. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005).
    [CrossRef] [PubMed]
  17. N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
    [CrossRef]
  18. C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
    [CrossRef]
  19. A. Lamas-Linares, C. Kurtsiefer, “Breaking a quantum key distribution system through a timing side channel,” Opt. Express 15, 9388–9393 (2007).
    [CrossRef] [PubMed]
  20. Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum key distribution systems,” Phys. Rev. A, 78, 042333 (2008).
    [CrossRef]
  21. L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
    [CrossRef]
  22. L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4, 686–689 (2010).
    [CrossRef]
  23. Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
    [CrossRef]
  24. L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
    [CrossRef]
  25. C. Bennett, G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proceedings of IEEE International Conference on Computers Systems and Signal Processing, 175 (1984).
  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). Note that we have corrected a mistake present in Eq. (17).
    [CrossRef]
  27. Note that a pulse does not necessarily contain one single photon. In particular, when considering attenuated light pulses, the number of photons in a pulse will, for example, follow the Poissonian distribution.
  28. F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
    [CrossRef]
  29. To the best of our knowledge, this assumption correctly describes all existing experimental implementations. See section 5 for more information.
  30. Note that this approximation is, in general, not correct. However, in order to obtain the best performance from a QKD implementation, the noise level should be as low as possible, i.e. Pn∼ 0.
  31. The separation of photons into genuine qubit photons and background photons is somewhat artificial – as a matter of fact, there is no way to distinguish background photons from real photons. As already stated in section 4.1, the distinction is motivated by the need to write down a general expression for all emitted single-photon qubit states using parameters that can be characterized directly through experiments (these measurements are further described below).
  32. D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
    [CrossRef]
  33. C. K. Hong, Z. Y. Ou, L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
    [CrossRef] [PubMed]
  34. L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A, 28, 929 (1983).
    [CrossRef]
  35. K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).
  36. M. Sasaki et al., “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express, 19, 10387–10409 (2011).
    [CrossRef] [PubMed]
  37. 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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
    [CrossRef]
  38. For instance, EOSpace sells intensity modulators with 50 dB extinction ratio.

2013 (6)

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013). Note that we have corrected a mistake present in Eq. (17).
[CrossRef]

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

2012 (2)

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

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

2011 (2)

L. Masanes, S. Pironio, A. Acín, “Secure device-independent quantum key distribution with causally independent measurement devices,” Nat. Commun. 2, 238 (2011).
[CrossRef] [PubMed]

M. Sasaki et al., “Field test of quantum key distribution in the Tokyo QKD network,” Opt. Express, 19, 10387–10409 (2011).
[CrossRef] [PubMed]

2010 (5)

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

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

Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

2009 (2)

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

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

2008 (1)

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

2007 (3)

A. Lamas-Linares, C. Kurtsiefer, “Breaking a quantum key distribution system through a timing side channel,” Opt. Express 15, 9388–9393 (2007).
[CrossRef] [PubMed]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

2006 (1)

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

2005 (2)

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

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

2003 (1)

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

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

2001 (1)

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

2000 (1)

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

1987 (1)

C. K. Hong, Z. Y. Ou, L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

1983 (1)

L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A, 28, 929 (1983).
[CrossRef]

Acín, A.

L. Masanes, S. Pironio, A. Acín, “Secure device-independent quantum key distribution with causally independent measurement devices,” Nat. Commun. 2, 238 (2011).
[CrossRef] [PubMed]

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Bechmann-Pasquinucci, H.

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

Bennett, C.

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

Brassard, G.

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

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

Braunstein, S. L.

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

Cerf, N. J.

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

Chan, P.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

Chen, C.

Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, 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, K.

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

Chen, T.-Y.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Cui, K.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Curty, M.

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[CrossRef]

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

da Silva, T. F.

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

Dixon, A.

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

Dušek, M.

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

Dynes, J.

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

Dynes, J. F.

Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
[CrossRef]

Elser, D.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

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

Fasel, S.

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

Fejer, M. M.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Fung, C.-H. F.

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

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).

Fürst, M.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Gerrits, T.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Gisin, N.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Gray, S.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Harrington, S.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

Hwang, W.

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

Kraus, B.

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

Kurtsiefer, C.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

A. Lamas-Linares, C. Kurtsiefer, “Breaking a quantum key distribution system through a timing side channel,” Opt. Express 15, 9388–9393 (2007).
[CrossRef] [PubMed]

Lamas-Linares, A.

Li, L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Liang, H.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Liao, Z.

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Lita, A. E.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Liu, N.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Liu, Y.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Lo, H.-K.

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[CrossRef]

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

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

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

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

K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Lucio-Martinez, I.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

Lütkenhaus, N.

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

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

Lydersen, L.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

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

Ma, X.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

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

Makarov, V.

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

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

Mandel, L.

C. K. Hong, Z. Y. Ou, L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A, 28, 929 (1983).
[CrossRef]

Marsili, F.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Masanes, L.

L. Masanes, S. Pironio, A. Acín, “Secure device-independent quantum key distribution with causally independent measurement devices,” Nat. Commun. 2, 238 (2011).
[CrossRef] [PubMed]

Mirin, R. P.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Mor, T.

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

Nam, S. W.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

Pan, J.-W.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Peev, M.

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

Pelc, J. S.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Perdigues, J.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Pirandola, S.

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

Pironio, S.

L. Masanes, S. Pironio, A. Acín, “Secure device-independent quantum key distribution with causally independent measurement devices,” Nat. Commun. 2, 238 (2011).
[CrossRef] [PubMed]

Qi, B.

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[CrossRef]

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

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

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Qian, L.

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Rarity, J.

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Rarity, J. G.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Ribordy, G.

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Rubenok, A.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

Sanders, B.

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

Sasaki, M.

Scarani, V.

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

Scheidl, T.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Schmitt-Manderbach, T.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Sharpe, A. W.

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

Shaw, M. D.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Shentu, G.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Shields, A.

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

Shields, A. J.

Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
[CrossRef]

Skaar, J.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

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

Slater, J. A.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

Sodnik, Z.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Stefanov, A.

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Stern, J. A.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Stucki, D.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Tamaki, K.

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).

Tang, Z.

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Temporão, G. P.

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

Ten, S.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Thew, R. T.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Tiefenbacher, F.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Tittel, W.

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[CrossRef] [PubMed]

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Towery, C. R.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Ursin, R.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Vannel, F.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Vayshenker, I.

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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Verma, V. 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, S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).
[CrossRef]

Vitoreti, D.

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

von der Weid, J. P.

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

Walenta, N.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Wall, T.

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Wang, J.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Wang, L.-J.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Wang, X.

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

Wang, X.-B.

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013). Note that we have corrected a mistake present in Eq. (17).
[CrossRef]

Weier, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Weinfurter, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Wiechers, C.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

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

Wittmann, C.

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

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Thermal blinding of gated detectors in quantum cryptography,” Opt. Express 18, 27938–27954 (2010).
[CrossRef]

Xavier, G. B.

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

Xu, F.

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[CrossRef]

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Yin, H.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Yuan, Z. L.

Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
[CrossRef]

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

Zbinden, H.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

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[CrossRef]

Zeilinger, A.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

Zhang, Q.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, Q. Zhang, J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[CrossRef] [PubMed]

Zhao, Y.

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

Appl. Phys. Lett. (1)

A. Dixon, Z. L. Yuan, J. Dynes, A. W. Sharpe, A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. 96, 161102 (2010).
[CrossRef]

J. Mod. Opt. (1)

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. Rarity, T. Wall, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,” J. Mod. Opt. 48, 1967–1981 (2001).
[CrossRef]

Nat. Commun. (1)

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[CrossRef] [PubMed]

Nat. Photonics (4)

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[CrossRef]

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

Z. L. Yuan, J. F. Dynes, A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800–801 (2010).
[CrossRef]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010).
[CrossRef]

New J. Phys. (2)

F. Xu, M. Curty, B. Qi, H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[CrossRef]

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, S. Ten, “High rate, long-distance quantum key distribution over 250 km of ultra low loss fibres,” New J. Phys. 11, 075003 (2009).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (5)

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

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73, 022320 (2006).
[CrossRef]

C.-H. F. Fung, B. Qi, K. Tamaki, H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” Phys. Rev. A 75, 032314 (2007).
[CrossRef]

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[CrossRef]

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013). Note that we have corrected a mistake present in Eq. (17).
[CrossRef]

Phys. Rev. A. (1)

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement device independent QKD using polarization qubits,” Phys. Rev. A. 88, 052303 (2013).
[CrossRef]

Phys. Rev. Lett. (10)

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[CrossRef]

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[CrossRef] [PubMed]

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

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144 km,” Phys. Rev. Lett., 98, 010504 (2007).
[CrossRef] [PubMed]

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

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

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

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[CrossRef]

Other (8)

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” arXiv:1306.6134 [quant-ph].

Note that a pulse does not necessarily contain one single photon. In particular, when considering attenuated light pulses, the number of photons in a pulse will, for example, follow the Poissonian distribution.

K. Tamaki, H.-K. Lo, C.-H. F. Fung, B. Qi, “Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw,” arxiv:1111.3413v4 (2013).

For instance, EOSpace sells intensity modulators with 50 dB extinction ratio.

To the best of our knowledge, this assumption correctly describes all existing experimental implementations. See section 5 for more information.

Note that this approximation is, in general, not correct. However, in order to obtain the best performance from a QKD implementation, the noise level should be as low as possible, i.e. Pn∼ 0.

The separation of photons into genuine qubit photons and background photons is somewhat artificial – as a matter of fact, there is no way to distinguish background photons from real photons. As already stated in section 4.1, the distinction is motivated by the need to write down a general expression for all emitted single-photon qubit states using parameters that can be characterized directly through experiments (these measurements are further described below).

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

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Figures (7)

Fig. 1
Fig. 1

Schematics for MDI-QKD. Charlie facilitates the key distribution between Alice and Bob without being able to learn the secret key.

Fig. 2
Fig. 2

Time-bin qubits are created at Alice’s and Bob’s through a CW laser (LD), attenuator (ATT), and frequency shifter (FS) and temperature-controlled intensity (IM) and phase (PM) modulator. The projective measurements are done at Charlie’s via a beam splitter (BS) and two single photon detectors (SPDs).

Fig. 3
Fig. 3

Sketch (not to scale) of the probability density p(t) for a detection event to occur as a function of time within one gate. Detection events can arise from a photon within an optical pulse (depicted here as a pulse in the late temporal mode), or be due to optical background, a dark count, or afterpulsing. Also shown are the 400 ps wide time-bins. Within the early time-bin only optical background, dark counts and afterpulsing give rise to detection events in this case. Note that the width of the temporal mode exceeds the widths of the time-bins.

Fig. 4
Fig. 4

Afterpulse probability per time-bin as a function of the average number of photons arriving at the detector per gate.

Fig. 5
Fig. 5

Modelled and measured results. Figure a) shows the plot for the error rates in the z-basis (green band) and in the x-basis (blue band) as a function of the mean photon number per pulse sent by Alice (μ) and Bob (σ) multiplied by the channel transmissions (tA and tB). Figure b) shows the plot of the gains as a function of µσtAtB. The z-basis is shown in green and the x-basis is shown in blue. For both figures the results of the measurements done in the laboratory are shown with squares (blue or green) and the measurements done over deployed fiber are shown with diamond (red and purple). The difference in gains and error rates in the x- and the z-basis, respectively is due to the fact that, in the case in which one party sends a laser pulse containing more than one photon and the other party sends zero photons, projections onto the |ψ〉 Bell state can only occur if both pulses encode qubits belonging to the x-basis. The Bell state projection cannot occur if both prepare qubits belonging to the z-basis (we ignore detector noise for the sake of this argument). This causes increased gain for the x-basis and, due to an error rate of 50% associated with these projections, also an increased error rate for the x-basis.

Fig. 6
Fig. 6

a) Optimum signal state intensity, τs, and b) corresponding secret key rate as a function of total loss in dB. The secondary axis shows distances assuming typical loss of 0.2 dB/km in optical fiber without splices. The optimum values for μs for small loss have to be taken with caution as in this regime the model needs to be expanded to higher photon number terms.

Fig. 7
Fig. 7

a) Optimum signal state intensity, τs, and b) corresponding secret key rate as a function of total loss in dB. The secondary axis shows distances assuming typical loss of 0.2 dB/km in optical fiber without splices. The optimum values for μs for small loss, are not shown as the model needs to be expanded to higher photon number terms in this regime.

Tables (2)

Tables Icon

Table 2 Measured error rates, e μ σ x , z, and gains, Q μ σ x , z, for different mean photon numbers, μ and σ (where μ = σ), lengths of fiber connecting Alice and Charlie, and Charlie and Bob, A and B, respectively, and total transmission loss, l. The last set of data details real-world measurements using deployed fiber. Uncertainties are calculated using Poissonian detection statistics.

Tables Icon

Table 1 Experimentally established values for all parameters required to describe the generated quantum states, as defined in Eq. (2), as well as two-photon interference parameters and detector properties.

Equations (27)

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S = Q 11 z ( 1 h 2 ( e 11 x ) ) Q μ σ z f h 2 ( e μ σ z ) ,
| ψ = 1 1 + 2 b x , z ( m x , z + b x , z | 0 + e i ϕ x , z 1 m x , z + b x , z | 1 )
P ( | ψ | 0 photons , in ) = P ( | ψ | 0 photons , out ) = 2 P n 2 ,
P ( | ψ | 1 photons , in ) = η P n + ( 1 η ) P ( | ψ | 0 photons , out ) ,
P ( | ψ | 2 photons , 1 spatial mode , out ) = ( 1 ( 1 η ) 2 ) P n + ( 1 η ) 2 P ( | ψ | 0 photons , out ) .
P ( | ψ | 2 photons , 2 spatial mode , 1 temporal mode , out ) = 2 η ( 1 η ) P n + ( 1 η ) 2 P ( | ψ | 0 photons , out ) .
P ( | ψ | 2 photons , 2 spatial mode , 2 temporal mode , out ) = η 2 + 2 η ( 1 η ) P n + ( 1 η ) 2 P ( | ψ | 0 photons , out ) .
p x , z ( 0 , 0 ) ( m 1 x , z + b 1 x , z ) ( m 2 x , z + b 2 x , z ) p x , z ( 0 , 1 ) ( m 1 x , z + b 1 x , z ) ( 1 m 2 x , z + b 2 x , z ) p x , z ( 1 , 0 ) ( 1 m 1 x , z + b 1 x , z ) ( m 2 x , z + b 2 x , z ) p x , z ( 1 , 1 ) ( 1 m 1 x , z + b 1 x , z ) ( 1 m 2 x , z + b 2 x , z ) b norm x , z 1 + 2 b 1 x , z + 2 b 2 x , z + 4 b 1 x , z b 2 x , z
| ψ input = ( 1 1 + 2 b x , z ( m x , z + b x , z a ^ ( 0 ) + e i ϕ x , z 1 m x , z + b x , z a ^ ( 1 ) ) ) 2 | vac ,
P ( | ψ | 2 photons , 1 spatial mode , in ) = 1 2 P ( | ψ | 2 photons , 1 spatial mode , out ) + A × P ( | ψ | 2 photons , 2 spatial modes , 1 temporal mode , out ) + B × P ( | ψ | 2 photons , 2 spatial modes , 2 temporal modes , out ) .
| ψ input = 1 1 + 2 b 1 x , z ( m 1 x , z + b 1 x , z a ^ ( 0 ) + e i ϕ 1 x , z 1 m 1 x , z + b 1 x , z a ^ ( 1 ) ) 1 1 + 2 b 2 x , z ( m 2 x , z + b 2 x , z b ^ ( 0 ) + e i ϕ 2 x , z 1 m 2 x , z + b 2 x , z b ^ ( 1 ) ) | vac ,
P ( | ψ | 2 photons , 2 spatial modes , non-interfering , in ) = P ( | ψ | 2 photons , 1 spatial mode , in ) P ( | ψ | 2 photons , non-interfering , in ) .
P ( | ψ | 2 photons , interfering , in ) = C × P ( | ψ | 2 photons , 1 spatial mode , out ) + D × P ( | ψ | 2 photons , 2 spatial modes , 2 temporal modes , out ) .
P ( | ψ | 2 photons , visibility V , in ) = V P ( | ψ | 2 photons , interfering , in ) + ( 1 V ) P ( | ψ | 2 photons , non-interfering , in ) .
P ( a , det ) = k = k dead ( γ × υ × ρ × P k )
γ = ( 1 μ avg ( μ , σ , t A , t B ) η gate ) k k dead
υ = ( 1 P d , gate ) k k dead
ρ = j = k dead k 1 1 α p ( 1 p ) j
P k = α p ( 1 p ) k
μ avg ( μ , σ , t A , t B ) = ( μ + b A ) t A + ( σ + b B ) t B 2 ,
P a , gate = ( μ avg ( μ , σ , t A , t B ) η gate + P d , gate + P a , gate ) P ( a , det ) .
P a , gate = ( μ avg ( μ , σ , t A , t B ) η gate + P d , gate ) P ( a , det ) 1 P ( a , det ) .
P a ( μ , σ , t A , t B ) = P a , gate P d P d , gate .
Q 11 x , z 𝔻 1 ( τ s ) 𝔻 2 ( τ s ) ( Q d d x , z Q 0 x , z ( τ d ) ) 𝔻 1 ( τ d ) 𝔻 2 ( τ d ) ( Q s s x , z Q 0 x , z ( τ s ) ) 𝔻 1 ( τ s ) 𝔻 1 ( τ d ) ( 𝔻 1 ( τ d ) 𝔻 2 ( τ s ) 𝔻 1 ( τ s ) 𝔻 2 ( τ d ) ) ,
Q 0 x , z ( τ d ) = 𝔻 0 ( τ d ) Q v d x , z + 𝔻 0 ( τ d ) Q d v x , z 𝔻 0 ( τ d ) 2 Q v v x , z ,
Q 0 x , z ( τ s ) = 𝔻 0 ( τ s ) Q v s x , z + 𝔻 0 ( τ s ) Q s v x , z 𝔻 0 ( τ s ) 2 Q v v x , z .
e 11 x e d d x Q d d x 𝔻 0 ( τ d ) e v d x Q v d x 𝔻 0 ( τ d ) e d v x Q d v x + 𝔻 0 ( τ d ) 2 e v v x Q v v x 𝔻 1 ( τ d ) 2 Q 11 x ,

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