Abstract

Characterization of photon statistics of a light source is one of the most basic tools in quantum optics. Existing methods rely on an implicit and unverifiable assumption that the source never emits too many photons to stay within the measuring range of the detectors. As a result, they fail to fulfill the demand arising from emerging applications of quantum information such as quantum cryptography. Here, we propose a characterization method using a conventional Hanbury-Brown-Twiss setup to produce rigorous bounds on emission probabilities of low photon numbers from an unknown source. As an application, we show that our characterization method can be used for a practical light source in a quantum key distribution protocol to forsake the commonly used a priori assumption without significant change in efficiency. Our versatile and flexible formula for rigorous bounds will make an essential contribution to the optics toolbox in the era of quantum information.

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

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2018 (5)

P. Obšil, L. Lachman, T. Pham, A. Lešundák, V. Hucl, M. Čížek, J. Hrabina, O. Číp, L. Slodička, and R. Filip, “Nonclassical light from large ensembles of trapped ions,” Phys. Rev. Lett. 120, 253602 (2018).
[Crossref]

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4, 4 (2018).
[Crossref]

O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, “Geometrical picture of photocounting measurements,” Phys. Rev. A 97, 023845 (2018).
[Crossref]

J. F. Dynes, M. Lucamarini, K. A. Patel, A. W. Sharpe, M. B. Ward, Z. L. Yuan, and A. J. Shields, “Testing the photon-number statistics of a quantum key distribution light source,” Opt. Express 26, 22733–22749 (2018).
[Crossref] [PubMed]

2017 (1)

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

2016 (2)

L. Lachman and R. Filip, “Quantum non-Gaussianity from a large ensemble of single photon emitters,” Opt. Express 24, 27352–27359 (2016).
[Crossref] [PubMed]

L. Lachman, L. Slodička, and R. Filip, “Nonclassical light from a large number of independent single-photon emitters,” Sci. Reports 6, 19760 (2016).
[Crossref]

2015 (1)

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

2014 (3)

G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (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]

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

2013 (3)

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9, 143–252 (2013).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[Crossref]

R. Filip and L. Lachman, “Hierarchy of feasible nonclassicality criteria for sources of photons,” Phys. Rev. A 88, 043827 (2013).
[Crossref]

2012 (3)

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

V. Dunjko, E. Kashefi, and A. Leverrier, “Blind quantum computing with weak coherent pulses,” Phys. Rev. Lett. 108, 200502 (2012).
[Crossref] [PubMed]

S. Buckley, K. Rivoire, and J. Vuckovic, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75, 126503 (2012).
[Crossref]

2009 (2)

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

X.-B. Wang, L. Yang, C.-Z. Peng, and J.-W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11, 075006 (2009).
[Crossref]

2008 (1)

Y. Zhao, B. Qi, and H.-K. Lo, “Quantum key distribution with an unknown and untrusted source,” Phys. Rev. A 77, 052327 (2008).
[Crossref]

2007 (3)

Q. Wang, X.-B. Wang, and G.-C. Guo, “Practical decoy-state method in quantum key distribution with a heralded single-photon source,” Phys. Rev. A 75, 012312 (2007).
[Crossref]

M. Fujiwara and M. Sasaki, “Direct measurement of photon number statistics at telecom wavelengths using a charge integration photon detector,” Appl. Opt. 46, 3069–3074 (2007).
[Crossref] [PubMed]

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

2006 (1)

T. Horikiri and T. Kobayashi, “Decoy state quantum key distribution with a photon number resolved heralded single photon source,” Phys. Rev. A 73, 032331 (2006).
[Crossref]

2005 (6)

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

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

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]

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
[Crossref]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
[Crossref]

2004 (2)

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

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

2003 (2)

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

J. Řeháček, Z. Hradil, O. Haderka, J. Peřina, and M. Hamar, “Multiple-photon resolving fiber-loop detector,” Phys. Rev. A 67, 061801 (2003).
[Crossref]

2001 (1)

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

1993 (1)

M. Koashi, K. Kono, T. Hirano, and M. Matsuoka, “Photon antibunching in pulsed squeezed light generated via parametric amplification,” Phys. Rev. Lett. 71, 1164–1167 (1993).
[Crossref] [PubMed]

1977 (1)

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[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]

1957 (1)

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. Royal Soc. Lond. A: Math. Phys. Eng. Sci. 242, 300–324 (1957).
[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]

Aaronson, S.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9, 143–252 (2013).
[Crossref]

Adachi, Y.

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

Agarwal, G. S.

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

Aichele, T.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

Andreoni, A.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Ansari, V.

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

Arkhipov, A.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9, 143–252 (2013).
[Crossref]

Bajcsy, M.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Bartlett, S. D.

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

Bechmann-Pasquinucci, H.

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

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

Bohmann, M.

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

Bondani, M.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Bramati, A.

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

Brecht, B.

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

Brida, G.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Buckley, S.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

S. Buckley, K. Rivoire, and J. Vuckovic, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75, 126503 (2012).
[Crossref]

Carbone, L.

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

Cavanna, A.

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

Cerf, N. J.

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

Chekhova, M.

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

Chen, K.

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

Choi, I.

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L. Lachman, L. Slodička, and R. Filip, “Nonclassical light from a large number of independent single-photon emitters,” Sci. Reports 6, 19760 (2016).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
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Y. Zhao, B. Qi, and H.-K. Lo, “Quantum key distribution with an unknown and untrusted source,” Phys. Rev. A 77, 052327 (2008).
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[Crossref] [PubMed]

M. Lucamarini, J. F. Dynes, I. Choi, M. B. Ward, B. Frohlich, Z. L. Yuan, and A. J. Shields, “Practical security of a quantum key distribution transmitter,” in Invited Talk 9648–41, SPIE Security + Defence, Toulouse, 2015 (unpublished), (2015).

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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect device,” Quant. Inf. Comp. 4, 325 (2004).

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A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

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X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005).
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H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
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H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
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M. Koashi, K. Kono, T. Hirano, and M. Matsuoka, “Photon antibunching in pulsed squeezed light generated via parametric amplification,” Phys. Rev. Lett. 71, 1164–1167 (1993).
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I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4, 4 (2018).
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I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4, 4 (2018).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
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A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

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G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
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[Crossref] [PubMed]

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Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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P. Obšil, L. Lachman, T. Pham, A. Lešundák, V. Hucl, M. Čížek, J. Hrabina, O. Číp, L. Slodička, and R. Filip, “Nonclassical light from large ensembles of trapped ions,” Phys. Rev. Lett. 120, 253602 (2018).
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A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

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

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Y. Zhao, B. Qi, and H.-K. Lo, “Quantum key distribution with an unknown and untrusted source,” Phys. Rev. A 77, 052327 (2008).
[Crossref]

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

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L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

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G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
[Crossref]

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J. Řeháček, Z. Hradil, O. Haderka, J. Peřina, and M. Hamar, “Multiple-photon resolving fiber-loop detector,” Phys. Rev. A 67, 061801 (2003).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
[Crossref]

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G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
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A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
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G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
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Sanders, B. C.

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

Sarmiento, T.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Sasaki, M.

Scarani, V.

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

Schiller, S.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

Semenov, A. A.

O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, “Geometrical picture of photocounting measurements,” Phys. Rev. A 97, 023845 (2018).
[Crossref]

Sharpe, A. W.

Shields, A. J.

J. F. Dynes, M. Lucamarini, K. A. Patel, A. W. Sharpe, M. B. Ward, Z. L. Yuan, and A. J. Shields, “Testing the photon-number statistics of a quantum key distribution light source,” Opt. Express 26, 22733–22749 (2018).
[Crossref] [PubMed]

M. Lucamarini, J. F. Dynes, I. Choi, M. B. Ward, B. Frohlich, Z. L. Yuan, and A. J. Shields, “Practical security of a quantum key distribution transmitter,” in Invited Talk 9648–41, SPIE Security + Defence, Toulouse, 2015 (unpublished), (2015).

Silberhorn, C.

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
[Crossref]

Slodicka, L.

P. Obšil, L. Lachman, T. Pham, A. Lešundák, V. Hucl, M. Čížek, J. Hrabina, O. Číp, L. Slodička, and R. Filip, “Nonclassical light from large ensembles of trapped ions,” Phys. Rev. Lett. 120, 253602 (2018).
[Crossref]

L. Lachman, L. Slodička, and R. Filip, “Nonclassical light from a large number of independent single-photon emitters,” Sci. Reports 6, 19760 (2016).
[Crossref]

Sperling, J.

O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, “Geometrical picture of photocounting measurements,” Phys. Rev. A 97, 023845 (2018).
[Crossref]

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

Stevens, M. J.

Stoklasa, B.

G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
[Crossref]

Straka, I.

I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4, 4 (2018).
[Crossref]

Twiss, R. Q.

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. Royal Soc. Lond. A: Math. Phys. Eng. Sci. 242, 300–324 (1957).
[Crossref]

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

Vogel, W.

O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, “Geometrical picture of photocounting measurements,” Phys. Rev. A 97, 023845 (2018).
[Crossref]

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

Vuckovic, J.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

S. Buckley, K. Rivoire, and J. Vuckovic, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75, 126503 (2012).
[Crossref]

Waks, E.

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, X.-B. Wang, and G.-C. Guo, “Practical decoy-state method in quantum key distribution with a heralded single-photon source,” Phys. Rev. A 75, 012312 (2007).
[Crossref]

Wang, X.-B.

X.-B. Wang, L. Yang, C.-Z. Peng, and J.-W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11, 075006 (2009).
[Crossref]

Q. Wang, X.-B. Wang, and G.-C. Guo, “Practical decoy-state method in quantum key distribution with a heralded single-photon source,” Phys. Rev. A 75, 012312 (2007).
[Crossref]

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

Ward, M. B.

J. F. Dynes, M. Lucamarini, K. A. Patel, A. W. Sharpe, M. B. Ward, Z. L. Yuan, and A. J. Shields, “Testing the photon-number statistics of a quantum key distribution light source,” Opt. Express 26, 22733–22749 (2018).
[Crossref] [PubMed]

M. Lucamarini, J. F. Dynes, I. Choi, M. B. Ward, B. Frohlich, Z. L. Yuan, and A. J. Shields, “Practical security of a quantum key distribution transmitter,” in Invited Talk 9648–41, SPIE Security + Defence, Toulouse, 2015 (unpublished), (2015).

Yamamoto, T.

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

Yamamoto, Y.

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

Yang, L.

X.-B. Wang, L. Yang, C.-Z. Peng, and J.-W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11, 075006 (2009).
[Crossref]

Yuan, Z. L.

J. F. Dynes, M. Lucamarini, K. A. Patel, A. W. Sharpe, M. B. Ward, Z. L. Yuan, and A. J. Shields, “Testing the photon-number statistics of a quantum key distribution light source,” Opt. Express 26, 22733–22749 (2018).
[Crossref] [PubMed]

M. Lucamarini, J. F. Dynes, I. Choi, M. B. Ward, B. Frohlich, Z. L. Yuan, and A. J. Shields, “Practical security of a quantum key distribution transmitter,” in Invited Talk 9648–41, SPIE Security + Defence, Toulouse, 2015 (unpublished), (2015).

Zambra, G.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Zhao, Y.

Y. Zhao, B. Qi, and H.-K. Lo, “Quantum key distribution with an unknown and untrusted source,” Phys. Rev. A 77, 052327 (2008).
[Crossref]

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

Appl. Opt. (1)

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)

X.-B. Wang, L. Yang, C.-Z. Peng, and J.-W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11, 075006 (2009).
[Crossref]

L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, “Multiphoton nonclassical light from clusters of single-photon emitters,” New J. Phys. 20, 073013 (2018).
[Crossref]

npj Quantum Inf. (1)

I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4, 4 (2018).
[Crossref]

Opt. Express (3)

Phys. Rev. (2)

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[Crossref]

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

Phys. Rev. A (13)

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

J. Řeháček, Z. Hradil, O. Haderka, J. Peřina, and M. Hamar, “Multiple-photon resolving fiber-loop detector,” Phys. Rev. A 67, 061801 (2003).
[Crossref]

G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Time-multiplexed measurements of nonclassical light at telecom wavelengths,” Phys. Rev. A 90, 042105 (2014).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[Crossref]

R. Filip and L. Lachman, “Hierarchy of feasible nonclassicality criteria for sources of photons,” Phys. Rev. A 88, 043827 (2013).
[Crossref]

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

T. Horikiri and T. Kobayashi, “Decoy state quantum key distribution with a photon number resolved heralded single photon source,” Phys. Rev. A 73, 032331 (2006).
[Crossref]

Q. Wang, X.-B. Wang, and G.-C. Guo, “Practical decoy-state method in quantum key distribution with a heralded single-photon source,” Phys. Rev. A 75, 012312 (2007).
[Crossref]

Y. Zhao, B. Qi, and H.-K. Lo, “Quantum key distribution with an unknown and untrusted source,” Phys. Rev. A 77, 052327 (2008).
[Crossref]

O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, “Geometrical picture of photocounting measurements,” Phys. Rev. A 97, 023845 (2018).
[Crossref]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[Crossref]

M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, “Direct calibration of click-counting detectors,” Phys. Rev. A 95, 033806 (2017).
[Crossref]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
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P. Obšil, L. Lachman, T. Pham, A. Lešundák, V. Hucl, M. Čížek, J. Hrabina, O. Číp, L. Slodička, and R. Filip, “Nonclassical light from large ensembles of trapped ions,” Phys. Rev. Lett. 120, 253602 (2018).
[Crossref]

V. Dunjko, E. Kashefi, and A. Leverrier, “Blind quantum computing with weak coherent pulses,” Phys. Rev. Lett. 108, 200502 (2012).
[Crossref] [PubMed]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting,” Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref] [PubMed]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of nonclassical photon statistics in parametric down-conversion,” Phys. Rev. Lett. 92, 113602 (2004).
[Crossref] [PubMed]

J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, “Uncovering quantum correlations with time-multiplexed click detection,” Phys. Rev. Lett. 115, 023601 (2015).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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]

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, “Simple and efficient quantum key distribution with parametric down-conversion,” Phys. Rev. Lett. 99, 180503 (2007).
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M. Koashi, K. Kono, T. Hirano, and M. Matsuoka, “Photon antibunching in pulsed squeezed light generated via parametric amplification,” Phys. Rev. Lett. 71, 1164–1167 (1993).
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Proc. Royal Soc. Lond. A: Math. Phys. Eng. Sci. (1)

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. Royal Soc. Lond. A: Math. Phys. Eng. Sci. 242, 300–324 (1957).
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Quant. Inf. Comp. (1)

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

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B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005).
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S. Buckley, K. Rivoire, and J. Vuckovic, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75, 126503 (2012).
[Crossref]

Rev. Mod. Phys. (1)

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

Sci. Reports (1)

L. Lachman, L. Slodička, and R. Filip, “Nonclassical light from a large number of independent single-photon emitters,” Sci. Reports 6, 19760 (2016).
[Crossref]

Theory Comput. (1)

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Theory Comput. 9, 143–252 (2013).
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Other (1)

M. Lucamarini, J. F. Dynes, I. Choi, M. B. Ward, B. Frohlich, Z. L. Yuan, and A. J. Shields, “Practical security of a quantum key distribution transmitter,” in Invited Talk 9648–41, SPIE Security + Defence, Toulouse, 2015 (unpublished), (2015).

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Figures (4)

Fig. 1
Fig. 1 An implementation of the D = 4 case of our characterization method. The overall detection efficiencies are given by η 1 = R 1 T 2 η 1 ( det ), η 2 = R 1 R 2 η 2 ( det ), η 3 = T 1 R 3 η 3 ( det ), and η 4 = T 1 T 3 η 4 ( det ), where T k   ( k = 1 , , 3 ) and R k   ( k = 1 , , 3 ) are transmittance and reflectance, respectively.
Fig. 2
Fig. 2 Comparison between the true values and the bounds. Each graph depicts the true value and the bounds from the characterization method in the case of D = 2 , 3 , and 4 with η = η i = 0.025 (regardless of D) when applied to an ideal Poissonian source with p n = e μ μ n / n ! where μ is the mean photon number. (a), p 0 U and p 0 L, (b), p 1 U and p 1 L, (c), p 2 U, p 2 L, and p 2 U, (d), p 3 U, p 3 L, and p 3 U.
Fig. 3
Fig. 3 Asymptotic secure key rates per pulse for the decoy-state BB84 protocol with various assumptions on the light source. The protocol uses pulses with three different intensities, termed signal, decoy, and vacuum. We assume that the vacuum pulses contain no photons. (a), An ideal Poissonian source with known distributions, p n = e μ μ n / n ! for the signal pulses and p n = e μ μ n / n ! for the decoy pulses. (b)-(d), Based on the characterization method with D = 4 , 3 , 2 detectors applied to the same source. For the characterization, we assume that all the overall detection efficiencies { η i } i have the same value η. We generated simulated values of { c obs , r } assuming η = η 0 : = 0.1 / D. For application of Theorem 1, we used all the values of η in the region [ 0.99 η 0 , 1.01 η 0 ], and adopted the worst-case value of R as the key rate. We assume that statistical errors in estimating c obs , r are negligible. For the protocol, we assume q = 0.8, q = 0.1, Y 0 = 10 8 and that the channel causes a constant error of 1% regardless of the transmission. The detection rate Q and the bit error rate E are modeled as Q / q = 1 exp   ( μ τ ) + Y 0 and Q E / q = 0.01 ( 1 exp   ( μ τ ) ) + 0.5 Y 0, where τ is the channel transmission. Q and E are defined similarly. The mean photon numbers are optimized for each value of τ under the condition μ = μ / 10.
Fig. 4
Fig. 4 Asymptotic secure key rates per pulse for the decoy-state BB84 protocol with a light source deviated from Poissonian. The photon number distribution for the signal pulse is a uniform mixture of Poissonian with mean μ ˜ over the range μ ˜ [ 0.7 μ , 1.3 μ ]. The distribution for the decoy pulse is the same except μ replaced by μ = μ / 10. All the other parameters are the same as those in Fig. 3. (a’), The key rate when the distributions of the signal and the decoy pulse are known. (b’)-(d’), Based on the characterization method with D = 4 , 3 , 2 detectors applied to the same source. The black curve (a) which is slightly above (a’) is the same as the curve (a) in Fig. 3, shown for helping the comparison.

Equations (68)

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c obs , r = c n , r : = j = 0 r ( 1 ) j ω r , j W I j ( 1 i W η i ) n ,
c obs , r = n = 0 p n c n , r   ( r = 1 , , D ) .
i W η i < i W η i   if  | W | < | W |
p n p n U : = { c obs d n ( S )   ( D n : even ) ( 4 ) c obs d n ( S )   ( D n : odd ) ( 5 )
p n p n L : = { c obs d n ( S )   ( D n : even ) ( 6 ) c obs d n ( S )   ( D n : odd ) ( 7 )
n = D p n p D U : = c obs d D ( S )
n = D p n p D L : = c obs d ( S ) .
p n > 1 D ( D n ) p D 1
n < g ( D 1 ) ( 0 ) g ( D ) ( 0 ) .
Δ p 0 U , L p 0 p 1 p 0 ( | Δ c obs , 1 | c obs , 1 + | Δ η | η ) ,
Δ p n U , L p n | Δ c obs , n | c obs , n + n | Δ η | η ( n 1 ) .
R = ( q p 1 L + q p 1 L ) Y 1 L ( 1 H ( e 1 U ) ) ( Q + Q ) H ( ( Q E + Q E ) / ( Q + Q ) ) .
Y 1 L = p 2 Q / q p 2 Q / q ( p 0 p 2 p 0 p 2 ) Y 0 p 1 p 2 p 1 p 2
e 1 U = ( Q E / q p 0 e 0 Y 0 ) / ( p 1 Y 1 L ) ,
Y 1 L = p 2 L Q / q p 2 U Q / q ( p 0 U p 2 L p 0 L p 2 U ) Y 0 p 1 U p 2 L p 1 L p 2 U p 2 L p D U p 1 U p 2 L p 1 L p 2 U
e 1 U = ( Q E / q p 0 L e 0 Y 0 ) / ( p 1 L Y 1 L ) .
c obs , W = Prob { i W C i } .
c obs , W = 1 Prob { i W C ¯ i } = 1 W : W W , | W | 1 ( 1 ) | W | + 1 Prob { i W C ¯ i } .
Prob { i W C ¯ i } = Tr ( χ ( ρ in ) i W | 0 0 | i ) .
Prob { i W C ¯ i } = ( 1 i W η i ) n .
c obs , W = W : W W ( 1 ) | W | ( 1 i W η i ) n .
| { W I r | W W } | = ( D | W | r | W | ) ,
( D r ) c obs , r = W : W I r c obs , W = W : | W | r ( D | W | r | W | ) ( 1 ) | W | ( 1 i W η i ) n ,
c obs , W = i W ( 1 e η i μ ) = W : W W ( 1 ) | W | exp  ( μ i W η i ) ,
( D r ) c obs , r = W : W I r c obs , W = W : | W | r ( D | W | r | W | ) ( 1 ) | W | exp  ( μ i W η i ) .
n μ n n ! c n , r = j = 0 r ( 1 ) j ω r , j W I j exp  ( μ μ i W η i ) .
f ( m ) : = z 0 + j = 1 D ( 1 ) j z j W I j e α W m ,
D β 1 D 0 f ( m ) = j = 1 D W I j ( 1 ) j ( β 1 α W ) α W z j e α W m .
p 0 L = 1 c ˜ obs , 1 + 2 ( 1 + ξ 2 , 1 ) η ( 1 η ) c ˜ obs , 2 ,
p 0 U = 1 c ˜ obs , 1 + [ 1 ( 1 ξ 2 , 1 ) η ] c ˜ obs , 2 ,
p 1 L = c ˜ obs , 1 [ 2 ( 1 ξ 2 , 1 ) η ] c ˜ obs , 2 ,
p 1 U = c ˜ obs , 1 2 ( 1 + ξ 2 , 1 ) η c ˜ obs , 2 ,
p 2 L = 2 ! ( 1 + ξ 2 , 1 ) η 2 c ˜ obs , 2 ,
p 2 U = c ˜ obs , 2 .
p 0 L = 1 c ˜ obs , 1 + [ 1 ( 1 2 ξ 2 , 1 ) η ] c ˜ obs , 2 [ 1 ( 3 3 ξ 3 , 2 / 2 ) η + ( 2 9 ξ 3 , 2 / 2 + 2 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 ,
p 0 U = 1 c ˜ obs , 1 + [ 1 ( 1 2 ξ 2 , 1 ) η ] c ˜ obs , 2 3 ( 1 + ξ 3 , 2 ) η ( 1 3 η + 2 ( 1 + ξ 2 , 1 ) η 2 ) c ˜ obs , 3 ,
p 1 L = c ˜ obs , 1 [ 2 ( 1 2 ξ 2 , 1 ) η ] c ˜ obs , 2 + 3 ( 1 + ξ 3 , 2 ) η ( 2 3 η ) c ˜ obs , 3 ,
p 1 U = c ˜ obs , 1 [ 2 ( 1 2 ξ 2 , 1 ) η ] c ˜ obs , 2 + [ 3 ( 6 3 ξ 3 , 2 ) η + ( 2 9 / 2 ξ 3 , 2 + 2 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 ,
p 2 L = c ˜ obs , 2 3 [ 1 ( 1 ξ 3 , 2 / 2 ) η ] c ˜ obs , 3 ,
p 2 U = c ˜ obs , 2 3 ( 1 + ξ 3 , 2 ) η c ˜ obs , 3 ,
p 3 L = 3 ! ( 1 + ξ 3 , 1 ) η 3 c ˜ obs , 3 ,
p 3 U = c ˜ obs , 3 .
p 0 L = 1 c ˜ obs , 1 + [ 1 ( 1 3 ξ 2 , 1 ) η ] c ˜ obs , 2 [ 1 ( 3 3 ξ 3 , 2 ) η + ( 2 12 ξ 3 , 2 + 6 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 + 4 ( 1 + ξ 4 , 3 ) η [ 1 6 η + ( 11 + 3 ξ 3 , 1 ) η 2 6 ( 1 + ξ 3 , 1 ) η 3 ] c ˜ obs , 4 ,
p 0 U = 1 c ˜ obs , 1 + [ 1 ( 1 3 ξ 2 , 1 ) η ] c ˜ obs , 2 [ 1 ( 3 3 ξ 3 , 2 ) η + ( 2 12 ξ 3 , 2 + 6 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 + [ 1 ( 6 2 ξ 4 , 3 ) η + ( 11 + 6 ξ 2 , 1 8 ξ 3 , 2 / 3 12 ξ 4 , 3 + 11 ξ 4 , 2 / 3 ) η 2 ( 6 + 24 ξ 2 , 1 32 ξ 3 , 2 / 3 16 ξ 4 , 3 + 44 ξ 4 , 2 / 3 6 ξ 4 , 1 ) η 3 ] c ˜ obs , 4 ,
p 1 L = c ˜ obs , 1 [ 2 ( 1 3 ξ 2 , 1 ) η ] c ˜ obs , 2 + [ 3 ( 6 6 ξ 3 , 2 ) η + ( 2 12 ξ 3 , 2 + 6 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 [ 4 ( 18 6 ξ 4 , 3 ) η + ( 22 + 12 ξ 2 , 1 16 ξ 3 , 2 / 3 24 ξ 4 , 3 + 22 ξ 4 , 2 / 3 ) η 2 ( 6 + 24 ξ 2 , 1 32 ξ 3 , 2 / 3 16 ξ 4 , 3 + 44 ξ 4 , 2 / 3 6 ξ 4 , 1 ) η 3 ] c ˜ obs , 4 ,
p 1 U = c ˜ obs , 1 [ 2 ( 1 3 ξ 2 , 1 ) η ] c ˜ obs , 2 + [ 3 ( 6 6 ξ 3 , 2 ) η + ( 2 12 ξ 3 , 2 + 6 ξ 3 , 1 ) η 2 ] c ˜ obs , 3 4 ( 1 + ξ 4 , 3 ) η [ 3 12 η + ( 11 + 3 ξ 3 , 1 ) η 2 ] c ˜ obs , 4 ,
p 2 L = c ˜ obs , 2 3 [ 1 ( 1 ξ 3 , 2 ) η ] c ˜ obs , 3 + 12 ( 1 + ξ 4 , 3 ) η ( 1 2 η ) c ˜ obs , 4 ,
p 2 U = c ˜ obs , 2 3 [ 1 ( 1 ξ 3 , 2 ) η ] c ˜ obs , 3 + [ 6 ( 18 6 ξ 4 , 3 ) η + ( 11 + 6 ξ 2 , 1 8 ξ 3 , 2 / 3 12 ξ 4 , 3 + 11 ξ 4 , 2 / 3 ) η 2 ] c ˜ obs , 4 ,
p 3 L = c ˜ obs , 3 [ 4 2 ( 3 ξ 4 , 3 ) η ] c ˜ obs , 4 ,
p 3 U = c ˜ obs , 3 4 ( 1 + ξ 4 , 3 ) η c ˜ obs , 4 ,
p 4 L = 4 ! ( 1 + ξ 4 , 1 ) η 4 c ˜ obs , 4 ,
p 4 U = c ˜ obs , 4 .
( n ) D : = n n ( n 1 ) ( n D + 1 ) p n ,
n = 0 D 1 c n , r p n ( S , D ) + p ( S , D ) = c obs , r
n = 0 D 1 c n , r p n ( S , D 1 ) = c obs , r
n = 0 D 1 η r c n , r ( p n ( S , D 1 ) p n ( S , D ) ) = η r c obs , D
p n ( S , D 1 ) p n ( S , D ) = O ( η ) ,
c obs d n ( S ) 0   ( 0 n D ) ,
c obs d n ( S ) 0   ( 0 n D 1 )
c obs d ( S ) 0 ,
c m d n ( S ) m ( m 1 ) ( m D ) m n ( 1 ) D n n ! ( D n ) ! .
c obs d n ( S ) p n m D + 1 m ( m 1 ) ( m D ) m n χ n p m
c obs d n ( S ) p n χ n χ D 1 ( p D 1 c obs d D 1 ( S ) ) = p n 1 D ( D n ) ( p D 1 c obs d D 1 ( S ) )
Q q n = 0 D 1 p n U Y n + p D U
Q q n = 0 D 1 p n L Y n .
p 2 L Q q p 2 U Q q n = 0 , 1 ( p 2 L p n U p 2 U p n L ) Y n + p 2 L p D U .
Q E q p 0 L Y 0 e 0 + p 1 L Y 1 e 1 ,
Y 1 L = ( Q q p 0 U Y 0 p D U ) / p 1 U .

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