Abstract

A commonly held tenet is that lasers well above threshold emit photons in a coherent state, which follow Poissonian statistics when measured in photon number. This feature is often exploited to build quantum-based random number generators or to derive the secure key rate of quantum key distribution systems. Hence the photon number distribution of the light source can directly impact the randomness and the security distilled from such devices. Here, we propose a method based on measuring correlation functions to experimentally characterize a light source’s photon statistics and use it in the estimation of a quantum key distribution system’s key rate. This promises to be a useful tool for the certification of quantum-related technologies.

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

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References

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2017 (1)

2016 (1)

K. Tamaki, M. Curty, and M. Lucamarini, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18, 065008 (2016).
[Crossref]

2015 (2)

2014 (6)

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
[Crossref]

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. W. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22, 1645–1654 (2014).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. J. Stevens, S. Glancy, S. W. Nam, and R. P. Mirin, “Third-order antibunching from an imperfect single-photon source,” Opt. Express 22, 3244–3260 (2014).
[Crossref] [PubMed]

M. Hayashi and R. Nakayama, “Security analysis of the decoy method with the bennett–brassard 1984 protocol for finite key lengths,” New J. Phys. 16, 063009 (2014).
[Crossref]

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, 374 (2014).
[Crossref]

2013 (2)

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Efficient decoy-state quantum key distribution with quantified security,” Opt. Express 21, 24550–24565 (2013).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

2011 (1)

2010 (3)

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. review. E, Stat. nonlinear, soft matter physics 81, 051137 (2010).
[Crossref]

B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. letters 35, 312–314 (2010).
[Crossref]

M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express 18, 13029–13037 (2010).
[Crossref] [PubMed]

2009 (1)

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

2008 (2)

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[Crossref]

V. Scarani and R. Renner, “Quantum cryptography with finite resources: Unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. 100, 200501 (2008).
[Crossref] [PubMed]

2007 (3)

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
[Crossref]

Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91, 041114 (2007).
[Crossref]

M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instruments 78, 045104 (2007).
[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]

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and a proof of its unconditional security,” J. Cryptol. 18, 133–165 (2005).
[Crossref]

2004 (1)

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
[Crossref]

2003 (1)

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

2002 (1)

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

2000 (2)

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
[Crossref]

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

1999 (1)

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
[Crossref]

1994 (1)

J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
[Crossref]

1991 (1)

Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

1966 (1)

A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. 16, 1169–1172 (1966).
[Crossref]

1963 (2)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

1934 (1)

C. J. Clopper and E. S. Pearson, “The use of confidence or fiducial limits illustrated in the case of the binomial,” Biometrika 26, 404–413 (1934).
[Crossref]

Abellán, C.

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
[Crossref]

Acín, A.

Amaya, W.

Anzolin, G.

Ardehali, M.

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and a proof of its unconditional security,” J. Cryptol. 18, 133–165 (2005).
[Crossref]

Armstrong, J. A.

A. W. Smith and J. A. Armstrong, “Laser photon counting distributions near threshold,” Phys. Rev. Lett. 16, 1169–1172 (1966).
[Crossref]

Avella, A.

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. (IEEE, New York, 1984), p. 175. (1984).

Berchera, I. R.

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. (IEEE, New York, 1984), p. 175. (1984).

Brida, G.

F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Capmany, J.

Chau, H. F.

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and a proof of its unconditional security,” J. Cryptol. 18, 133–165 (2005).
[Crossref]

Chen, K.

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

Chi, Y.-M.

B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. letters 35, 312–314 (2010).
[Crossref]

Clopper, C. J.

C. J. Clopper and E. S. Pearson, “The use of confidence or fiducial limits illustrated in the case of the binomial,” Biometrika 26, 404–413 (1934).
[Crossref]

Curty, M.

Degiovanni, I. P.

F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Dixon, A. R.

Dynes, J. F.

A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Efficient decoy-state quantum key distribution with quantified security,” Opt. Express 21, 24550–24565 (2013).
[Crossref] [PubMed]

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[Crossref]

M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Practical treatment of quantum bugs,” (SPIE, 2012), SPIE Proceedings, p. 85421K.

Ekert,

Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

Fattal, D.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
[Crossref]

Fröhlich, B.

Fujita, K.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
[Crossref]

Fujiwara, M.

Fürst, M.

Genovese, M.

F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Gisin, N.

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

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

Glancy, S.

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Goldschmidt, E. A.

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Guinnard, L.

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

Guinnard, O.

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

Guo, H.

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. review. E, Stat. nonlinear, soft matter physics 81, 051137 (2010).
[Crossref]

Hadfield, R. H.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. i. basic theory: The correlation between photons in coherent beams of radiation,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. (IEEE, New York), p. 175. 242, 300–324 (1957).

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M. Hayashi and R. Nakayama, “Security analysis of the decoy method with the bennett–brassard 1984 protocol for finite key lengths,” New J. Phys. 16, 063009 (2014).
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H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
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Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
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T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
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Jofre, M.

Kardynal, B. E.

Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91, 041114 (2007).
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Kawamura, S.

Kimura, H.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
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M. Kumazawa, T. Sasaki, and M. Koashi, “Rigorous calibration method for photon-number statistics,” eprint arXiv:1710.00457 (2017).

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F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
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Lim, C. C. W.

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, 374 (2014).
[Crossref]

Liu, Y.

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. review. E, Stat. nonlinear, soft matter physics 81, 051137 (2010).
[Crossref]

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B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. letters 35, 312–314 (2010).
[Crossref]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and a proof of its unconditional security,” J. Cryptol. 18, 133–165 (2005).
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Loudon, R.

R. Loudon, The quantum theory of light (Oxford University, Oxford, 2000), 3rd ed.

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K. Tamaki, M. Curty, and M. Lucamarini, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18, 065008 (2016).
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A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Efficient decoy-state quantum key distribution with quantified security,” Opt. Express 21, 24550–24565 (2013).
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M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Practical treatment of quantum bugs,” (SPIE, 2012), SPIE Proceedings, p. 85421K.

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H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
[Crossref]

Ma, X.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
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H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, 1995), 1st ed.
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Marangon, D.

Mayers, D.

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
[Crossref]

Migdall, A.

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
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Mitchell, M. W.

Miyatake, O.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
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Nakayama, R.

M. Hayashi and R. Nakayama, “Security analysis of the decoy method with the bennett–brassard 1984 protocol for finite key lengths,” New J. Phys. 16, 063009 (2014).
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Nam, S. W.

Nauerth, S.

Nie, Y.-Q.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
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Okamoto, A.

Owens, P.

J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
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Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
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Patel, K. A.

Pearson, E. S.

C. J. Clopper and E. S. Pearson, “The use of confidence or fiducial limits illustrated in the case of the binomial,” Biometrika 26, 404–413 (1934).
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Penty, R. V.

Peters, S.

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
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F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
[Crossref]

Plews, A.

A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

Polyakov, S. V.

F. Piacentini, M. P. Levi, A. Avella, M. López, S. Kück, S. V. Polyakov, I. P. Degiovanni, G. Brida, and M. Genovese, “Positive operator-valued measure reconstruction of a beam-splitter tree-based photon-number-resolving detector,” Opt. Lett. 40, 1548–1551 (2015).
[Crossref] [PubMed]

E. A. Goldschmidt, F. Piacentini, I. R. Berchera, S. V. Polyakov, S. Peters, S. Kück, G. Brida, I. P. Degiovanni, A. Migdall, and M. Genovese, “Mode reconstruction of a light field by multiphoton statistics,” Phys. Rev. A 88, 013822 (2013).
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Pruneri, V.

Qi, B.

B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. letters 35, 312–314 (2010).
[Crossref]

Qian, L.

B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. letters 35, 312–314 (2010).
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J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
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M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instruments 78, 045104 (2007).
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C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
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Sasaki, T.

M. Kumazawa, T. Sasaki, and M. Koashi, “Rigorous calibration method for photon-number statistics,” eprint arXiv:1710.00457 (2017).

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Scarani, V.

V. Scarani and R. Renner, “Quantum cryptography with finite resources: Unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. 100, 200501 (2008).
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Sharpe, A. W.

Shields, A. J.

A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Efficient decoy-state quantum key distribution with quantified security,” Opt. Express 21, 24550–24565 (2013).
[Crossref] [PubMed]

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[Crossref]

Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91, 041114 (2007).
[Crossref]

M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Practical treatment of quantum bugs,” (SPIE, 2012), SPIE Proceedings, p. 85421K.

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C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
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A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

Steinlechner, F.

Stevens, M. J.

Stipcevic, M.

M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instruments 78, 045104 (2007).
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Tajima, A.

Tam, S.

Tamaki, K.

K. Tamaki, M. Curty, and M. Lucamarini, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18, 065008 (2016).
[Crossref]

Tang, W.

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. review. E, Stat. nonlinear, soft matter physics 81, 051137 (2010).
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Tanizawa, Y.

Tapster, P. R.

J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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Tomita, A.

Torres, J. P.

Toyama, M.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
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Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. i. basic theory: The correlation between photons in coherent beams of radiation,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. (IEEE, New York), p. 175. 242, 300–324 (1957).

Vuckovic, J.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
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Waks, E.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
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Walenta, N.

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, 374 (2014).
[Crossref]

Wang, J.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
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H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. review. E, Stat. nonlinear, soft matter physics 81, 051137 (2010).
[Crossref]

Weier, H.

Weihs, G.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
[Crossref]

Weinfurter, H.

M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express 18, 13029–13037 (2010).
[Crossref] [PubMed]

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, 1995), 1st ed.
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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, 374 (2014).
[Crossref]

Yamamoto, Y.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, E. Waks, and Y. Yamamoto, “Submicrosecond correlations in photoluminescence from inas quantum dots,” Phys. Rev. B 69, 1110 (2004).
[Crossref]

Yoshino, K.-i.

Yoshizawa, Y.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
[Crossref]

Yuan, Z. L.

A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref] [PubMed]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Efficient decoy-state quantum key distribution with quantified security,” Opt. Express 21, 24550–24565 (2013).
[Crossref] [PubMed]

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[Crossref]

Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91, 041114 (2007).
[Crossref]

M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Practical treatment of quantum bugs,” (SPIE, 2012), SPIE Proceedings, p. 85421K.

Zbinden, H.

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, 374 (2014).
[Crossref]

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

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

Zeilinger, a.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and a. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instruments 71, 1675 (2000).
[Crossref]

Zhang, H.-F.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
[Crossref]

Zhang, J.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
[Crossref]

Zhang, Z.

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
[Crossref]

Appl. Phys. Lett. (4)

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[Crossref]

Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, “Practical and fast quantum random number generation based on photon arrival time relative to external reference,” Appl. Phys. Lett. 104, 051110 (2014).
[Crossref]

Z. L. Yuan, M. Lucamarini, J. F. Dynes, B. Fröhlich, A. Plews, and A. J. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104, 261112 (2014).
[Crossref]

Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91, 041114 (2007).
[Crossref]

Biometrika (1)

C. J. Clopper and E. S. Pearson, “The use of confidence or fiducial limits illustrated in the case of the binomial,” Biometrika 26, 404–413 (1934).
[Crossref]

Eur. Phys. J. D (1)

H. Inamori, N. Lütkenhaus, and D. Mayers, “Unconditional security of practical quantum key distribution,” Eur. Phys. J. D 41, 599–627 (2007).
[Crossref]

J. Cryptol. (1)

H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and a proof of its unconditional security,” J. Cryptol. 18, 133–165 (2005).
[Crossref]

J. Jpn. Soc. Comput. Stat. (1)

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
[Crossref]

J. Mod. Opt. (2)

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).

J. G. Rarity, P. Owens, and P. R. Tapster, “Quantum random-number generation and key sharing,” J. Mod. Opt. 41, 2435–2444 (1994).
[Crossref]

Nat. Photonics (1)

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

Nature (1)

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

New J. Phys. (2)

K. Tamaki, M. Curty, and M. Lucamarini, “Decoy-state quantum key distribution with a leaky source,” New J. Phys. 18, 065008 (2016).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic of photon number distributions in a train of optical pulses with average photon number μ = 1. (a) Poissonian source; (b) super-Poissonian source; (c) single-photon source. Each colored dot represents a single photon. Vertical lines identify the optical pulses in the same time slot.
Fig. 2
Fig. 2 Schematic of the experimental arrangement for measuring normalized correlation functions up to the fourth order. ECA: electronically controlled attenuator, SD-APD: self-differencing avalanche photodiode.
Fig. 3
Fig. 3 (a): Laser DC current vs optical output power from DFB laser diode. Above threshold, around 4.5 mA, there is a steep rise in output power. (b): g2 − 1 as a function of laser optical intensity I, normalized to threshold optical intensity I0 with I0 = 12.67μW. Points: Experimental data. Solid curve: fit of the non-linear oscillator model using I0 as a free parameter. Dashed curve: non-linear oscillator model with I0 = 5.38μW.
Fig. 4
Fig. 4 (a): Measured normalized correlation functions g2, g3 and g4 for a DC current of 4.0 mA. (b): m-th order gm − 1 as a function of the order m. Red bars: experimental data; blue bars: prediction of non-linear oscillator model when I0 = 5.38 μW; wine bars: prediction of non-linear oscillator model when I0 = 12.67 μW.
Fig. 5
Fig. 5 (a): Measured normalized correlation functions g2, g3 and g4 for a DC current of 6.5 mA. (b): n-th order normalized correlation functions as a function of the order n. Red bars: experimental data; blue bars: prediction of non-linear oscillator model when I0 = 5.38 μW; wine bars: prediction of non-linear oscillator model when I0 = 12.67 μW.
Fig. 6
Fig. 6 Bounds to the photon probabilities p0, p1, p2 for various experimental conditions. In the top right corner of each diagram, the correlation functions used in the estimation of the bounds and the mean photon number μ are indicated. Underlined (overlined) quantities are for lower (upper) bounds obtained from the experimental data given in Sec. 3.1.2, Eqs. (15)(17), for a confidence interval of 7 standard deviations. The superscript P refers to the ideal Poissonian distribution. Numerical Values. Poissonian: p n P = e μ μ n / n !, with μ = {0.42, 0.1}. Bounds – clockwise starting from top-left diagram: p 0 _ = { 0.580 ; 0.655 ; 0.905 ; 0.655 ; } p 0 ¯ = { 0.670 ; 0.669 ; 0.905 ; 0.659 ; } p 1 _ = { 0.2411 ; 0.2411 ; 0.0903 ; 0.2702 ; } p 1 ¯ = { 0.4203 ; 0.2826 ; 0.0906 ; 0.2826 ; } p 2 _ = { 0 ; 0.04973 ; 0.00446 ; 0.04973 ; } p 2 ¯ = { 0.08947 ; 0.08947 ; 0.00461 ; 0.06405 . }
Fig. 7
Fig. 7 Secure key rate of the efficient decoy-state BB84 protocol in the finite-size scenario, for the ideal case of a Poissonian source (black line) and for the real case where the correlation functions up to the fourth order have been measured. The values of the correlation functions have all been drawn from the experiments. The blue, green and wine lines correspond to the source described in Sec. 3.1.2, Eqs. (15)(17), whereas the red line is for the source in Sec. 3.1.1, Eqs. (12)(14). The bounds for this latter source, which displays a quasi-thermal distribution, are given in the inset, together with the Poisson probabilities for comparative purposes. The mean photon number for the signal and the decoy states in the protocol have been set to u = 0.42 and v = 0.02, respectively. The security parameter is < 10−10 and the finite-size sample is drawn from a minimum of 1.2 × 1012 initial pulses.

Equations (60)

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μ = n p n n < 1 .
g ( 2 ) ( 0 ) = a ^ a ^ a ^ a ^ / a ^ a ^ 2
g ( 3 ) ( 0 ) = a ^ a ^ a ^ a ^ a ^ a ^ / a ^ a ^ 3
g ( 4 ) ( 0 ) = a ^ a ^ a ^ a ^ a ^ a ^ a ^ a ^ / a ^ a ^ 4
g ( 2 ) [ 0 ] = a ^ k a ^ k a ^ k a ^ k / a ^ k a ^ k 2
g ( 3 ) [ 0 ] = a ^ k a ^ k a ^ k a ^ k a ^ k a ^ k / a ^ k a ^ k 3
g ( 4 ) [ 0 ] = a ^ k a ^ k a ^ k a ^ k a ^ k a ^ k a ^ k a ^ k / a ^ k a ^ k 4
ρ = n = 0 p n | n n | .
g 2 = n = 0 p n [ n ( n 1 ) ] / μ 2
g 3 = n = 0 p n [ n ( n 1 ) ( n 2 ) ] / μ 3
g 4 = n = 0 p n [ n ( n 1 ) ( n 2 ) ( n 3 ) ] / μ 4
g 2 = 1.6985 ± 0.0138
g 3 = 4.21 ± 0.13
g 4 = 17.11 ± 2.84
g 2 = 1.0041 ± 0.0039
g 3 = 1.0059 ± 0.0056
g 4 = 1.099 ± 0.049
R p u p Z 2 { p _ 1 y _ 1 , Z [ 1 h ( e ¯ 1 , X ) ] f Q Z h ( E Z ) Δ } .
minimize p ^ 1
subject to 1 = n = 0 p ^ n
g ^ 2 = n = 0 p ^ n n ( n 1 ) / μ 2
g ^ 3 = n = 0 p ^ n n ( n 1 ) ( n 2 ) / μ 3
g ^ 4 = n = 0 p ^ n n ( n 1 ) ( n 2 ) ( n 3 ) / μ 4
0 p ^ n 1
g 2 _ g ^ 2 g 2 ¯
g 3 _ g ^ 3 g 3 ¯
g 4 _ g ^ 4 g 4 ¯
g 2 _ = g 2 * γ 1 σ g 2 ; g 2 ¯ = g 2 * + γ 1 σ g 2
g 3 _ = g 3 * γ 1 σ g 3 ; g 3 ¯ = g 3 * + γ 1 σ g 3
g 4 _ = g 4 * γ 1 σ g 4 ; g 4 ¯ = g 4 * + γ 1 σ g 4
n = 0 N cut p n n = N cut p n .
1 n = 0 N cut p ^ n 1 2 .
n = 0 N cut p ^ n n μ / 2 ,
n = 0 N cut p ^ n n 2 ( μ 2 g ^ 2 + μ ) / 2 ,
n = 0 N cut p ^ n n 3 ( μ 3 g ^ 3 + 3 μ 2 g ^ 2 + μ ) / 2 ,
n = 0 N cut p ^ n n 4 ( μ 4 g ^ 4 + 6 μ 3 g ^ 3 + 7 μ 2 g ^ 2 + μ ) / 2 .
μ = n = 0 p ^ n n
= n = 0 N cut p ^ n n + n = N cut + 1 p ^ n n
n = 0 N cut p ^ n n + ( N cut + 1 ) n = N cut + 1 p ^ n .
μ ( N cut + 1 ) ( 1 n = 0 N cut p ^ n ) n = 0 N cut p ^ n n
μ 2 g ^ 2 + μ ( N cut + 1 ) ( μ n = 0 N cut p ^ n n ) n = 0 N cut p ^ n n 2
μ 3 g ^ 3 + 3 μ 2 g ^ 2 + μ ( N cut + 1 ) ( μ 2 g ^ 2 + μ n = 0 N cut p ^ n n 2 ) n = 0 N cut p ^ n n 3
μ 4 g ^ 4 + 6 μ 3 g ^ 3 + 7 μ 2 g ^ 2 + μ ( N cut + 1 ) ( μ 3 g ^ 3 + 3 μ 2 g ^ 2 + μ n = 0 N cut p ^ n n 3 ) n = 0 N cut p ^ n n 4
minimize p ^ 1
subject to ( 32 ) ( 36 )
( 40 ) ( 43 )
( 24 ) ( 27 ) .
minimize y ^ 1
subject to 0 y ^ n 1
Y ( u ) = n = 0 p ^ n y ^ n
Y ( v ) = n = 0 n e v v n n ! y ^ n
Y ( w ) = n = 0 e w w n n ! y ^ n
p _ n p ^ n p ¯ n ,
minimize y ^ 1
subject to 0 y ^ n 1
Y ¯ ( u ) n = 0 N cut p _ n ( μ ) y ^ n
Y _ ( u ) n = 0 N cut p ¯ n ( μ ) y ^ n + Γ ¯ μ
Γ ¯ μ = 1 n = 0 N cut p _ n ( μ ) ,
e ¯ 1 = b ¯ 1 y _ 1 ,
e ¯ 1 = B ¯ ( u ) p _ 0 y _ 0 e _ 0 p _ 1 y _ 1 .

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