Abstract

Excess noise induced by the phase drifts is a serious impairment for the continuous-variable quantum key distribution with locally generated local oscillator scheme, which is recently proposed to avoid the side channel attacks due to the transmitted local oscillator. Theoretical and experimental studies on the phase estimation have been widely reported, while two frequency-locked laser sources are indispensable to achieve quantum coherent detection. Moreover, the self-referenced phase estimation scheme requires to propagate the strong reference pulse through optical fiber, which opens a security loophole through the manipulation of the reference pulse amplitude. Based on the theoretical security and Bayes’ theorem, we propose a phase estimation protocol, which does not require propagating the strong reference pulse for performing phase estimation. Compared to the other related work, the protocol can avoid the security problem caused by strong reference pulse. Moreover, this algorithm is an iterative progress for each of experiment to obtain the phase estimation and its uncertainty. We hope the proposed scheme could further promote the performance of continuous-variable quantum key distribution.

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

Full Article  |  PDF Article
OSA Recommended Articles
Reference pulse attack on continuous variable quantum key distribution with local local oscillator under trusted phase noise

Shengjun Ren, Rupesh Kumar, Adrian Wonfor, Xinke Tang, Richard Penty, and Ian White
J. Opt. Soc. Am. B 36(3) B7-B15 (2019)

Practical security of the continuous-variable quantum key distribution with real local oscillators under phase attack

Biao Huang, Yongmei Huang, and Zhenming Peng
Opt. Express 27(15) 20621-20631 (2019)

Dual-phase-modulated plug-and-play measurement-device-independent continuous-variable quantum key distribution

Qin Liao, Yijun Wang, Duan Huang, and Ying Guo
Opt. Express 26(16) 19907-19920 (2018)

References

  • View by:
  • |
  • |
  • |

  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 (2009).
    [Crossref]
  2. H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
    [Crossref]
  3. L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
    [Crossref]
  4. C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
    [Crossref]
  5. S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
    [Crossref]
  6. Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
    [Crossref]
  7. F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
    [Crossref]
  8. F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
    [Crossref] [PubMed]
  9. A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
    [Crossref] [PubMed]
  10. Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
    [Crossref] [PubMed]
  11. F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
    [Crossref] [PubMed]
  12. P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
    [Crossref]
  13. J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
    [Crossref]
  14. P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
    [Crossref]
  15. J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
    [Crossref]
  16. X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
    [Crossref]
  17. D. Huang, P. Huang, D. Lin, C. Wang, and G. Zeng, “High-speed continuous-variable quantum key distribution without sending a local oscillator,” Opt. Lett. 40(16), 3695–3698 (2015).
    [Crossref] [PubMed]
  18. B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).
  19. D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).
  20. H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
    [Crossref]
  21. E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
    [Crossref]
  22. S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).
  23. K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).
  24. H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
    [Crossref]
  25. B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).
  26. N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
    [Crossref]
  27. F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).
  28. M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
    [Crossref]
  29. V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
    [Crossref]
  30. L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
    [Crossref]
  31. L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
    [Crossref]
  32. Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).
  33. M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
    [Crossref]
  34. M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)
  35. T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
    [Crossref]
  36. F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).
  37. P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
    [Crossref]
  38. E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
    [Crossref]
  39. S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
    [Crossref] [PubMed]
  40. N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
    [Crossref]
  41. B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
    [Crossref]
  42. P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
    [Crossref]
  43. C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).
  44. N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).
  45. C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
    [Crossref]
  46. F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
    [Crossref]

2018 (1)

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

2017 (1)

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

2015 (7)

B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
[Crossref]

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

D. Huang, P. Huang, D. Lin, C. Wang, and G. Zeng, “High-speed continuous-variable quantum key distribution without sending a local oscillator,” Opt. Lett. 40(16), 3695–3698 (2015).
[Crossref] [PubMed]

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

2014 (2)

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

2013 (5)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

2012 (6)

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
[Crossref]

2011 (2)

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

2010 (3)

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
[Crossref]

2009 (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 (2009).
[Crossref]

2008 (1)

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

2007 (1)

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
[Crossref]

2006 (3)

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

2005 (2)

L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

2003 (2)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).

2002 (2)

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref] [PubMed]

2001 (1)

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

1999 (1)

E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
[Crossref]

1980 (1)

F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
[Crossref]

1973 (1)

P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
[Crossref]

1967 (1)

F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).

Assche, G. V.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[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 (2009).
[Crossref]

Bloch, E.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Bobrek, M.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

Bowerman, P. N.

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

Brif, C.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Brouri, R.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

Camacho, R.M.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
[Crossref]

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

Cerf, Nicolas J.

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

Chen, C.

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Chen, H.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Chen, W.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Chong, S. K.

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

Coldren, L.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Coles, P. J.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Consul, P. C.

P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
[Crossref]

Cory, D. G.

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

Curty, M.

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

Daneshgaran, F.

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

Delgado, M. T.

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

Diamanti, E.

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

Dridi, C.

C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).

Dušek, 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 (2009).
[Crossref]

Fang, J.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

Ferrie, C.

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

Freedman, M.

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

Fung, C. H. F.

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Garcá-Patrón, R.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

García-Patrón, Raúl

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

Gentile, A. A.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Gessner, M.

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

Giovannetti, V.

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
[Crossref]

Granade, C.

N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

Grangier, P.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref] [PubMed]

Grangier, Ph.

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

Grice, W.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

Griffith, Z.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Grosshans, F.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref] [PubMed]

Guo, G. C.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Haight, F. A.

F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).

Hall, M. J. W.

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

Han, Z. F.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Harbin,

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

Hastings, M. B.

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

He, G.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

Huang, D.

Huang, J. Z.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Huang, P.

Hwang, T.

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

Hyllus, P.

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

Ip, E.

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
[Crossref]

Jain, G. C.

P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
[Crossref]

Jiang, M. S.

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

Jing, M. A.

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

Johansson, L.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Jouguet, P.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

Kahn, J. M.

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
[Crossref]

Kumar, R.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Kunz-Jacques, S.

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

Lance, A. M.

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

Le, C. T.

C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
[Crossref]

Leverrier, A.

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

Li, H. W.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Li, W.

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

Li, Y.

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

Liang, L. M.

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

Lin, D.

Lin, J.

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Lo, H.

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Lo, H. K.

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

Lougovski, P.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

Lu, M.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Lütkenhaus, N.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

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 (2009).
[Crossref]

Ma, X. C.

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

Maccone, L.

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
[Crossref]

Megill, N. D

N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).

Mondin, M.

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

Nazarathy, M.

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

Nolty, R. G.

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

Obrien, J. L.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Ortlepp, T.

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

Paesani, S.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Park, H.-C.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Pavicic, M.

N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).

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 (2009).
[Crossref]

Penty, R.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Peter, V. L.

L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Pezzè, L.

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
[Crossref]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Pooser, R.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

Qi, B.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Reed, T.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Ren, S.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Rodwell, M.

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Samaniego, F. J.

F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
[Crossref]

Samuel, L. B.

L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Santagati, R.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Sarovar, M.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

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 (2009).
[Crossref]

Scheuer, E. M.

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Sharma, V.

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

Simony, E.

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

Smerzi, A.

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

Soh, D. B. S.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Sun, S. H.

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

Svore, K. M.

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

Symul, T.

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

Tamaki, K.

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

Tang, C.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Tang, X.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Tew, D. P.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Thompson, M. G.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Toepfer, H.

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

Tseng, H. Y.

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

Tsionas, E. G.

E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
[Crossref]

Uhlmann, H. F.

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

Urayama, J.

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Vincent, B. T.

B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
[Crossref]

Wang, C.

Wang, J.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

Wang, S.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

wang, T. H.

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

Weedbrook, C.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Wenger, J.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

White, I.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Wiebe, N.

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

Wiseman, H. M.

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

Wonfor, A.

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

Xu, B.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Xu, F.

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

Yadin, Y.

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

Yin, Z. Q.

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Yu, Z. G.

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

Zeng, G.

Zhang, H.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

Zhang, W.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Zhao, Y.

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Zhu, F.

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Applied Physics Letters (1)

T. Ortlepp, H. Toepfer, and H. F. Uhlmann, “Minimization of noise-induced bit error rate in a high Tc superconducting dc/single flux quantum converter,” Applied Physics Letters 78, 1279–1281 (2001).
[Crossref]

Biometrical J. (1)

C. T. Le, “Analysis of nested designs with binomially-distributed outcome variables,” Biometrical J.,  29, 153–161 (2010).
[Crossref]

Communications in Statistics (1)

E. G. Tsionas, “Bayesian analysis of the multivariate poisson distribution,” Communications in Statistics 28, 431–451 (1999).
[Crossref]

Comput. Econ. (1)

C. Dridi, “A short note on the numerical approximation of the standard normal cumulative distribution and its inverse,” Comput. Econ. 9, 77–81 (2003).

Entropy (1)

Y. Li, L. Pezzè, M. Gessner, W. Li, and A. Smerzi, “Frequentist and bayesian quantum phase estimation,” Entropy 20, 628 (2018).

IEEE J. Lightw. Technol. (1)

E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” IEEE J. Lightw. Technol. 25, 2675 (2007).
[Crossref]

IEEE Trans. Rel. (1)

P. N. Bowerman, R. G. Nolty, and E. M. Scheuer, “Calculation of the poisson cumulative distribution function (reliability applications),” IEEE Trans. Rel. 39, 158–161 (2002).
[Crossref]

J. Math. Psychol. (1)

B. T. Vincent, “A tutorial on Bayesian models of perception. Journal of Mathematical Psychology,” J. Math. Psychol. 66, 103–114 (2015).
[Crossref]

Journal of Lightwave Technology (1)

M. Nazarathy, E. Simony, and Y. Yadin, “Analytical evaluation of bit error rates for hard detection of optical differential phase amplitude shift keying (DPASK),” Journal of Lightwave Technology 24, 2248–2260 (2006).
[Crossref]

Journal of the Royal Statistical Society (1)

F. A. Haight, “Handbook of poisson distribution,” Journal of the Royal Statistical Society 18, 478 (1967).

Nat. Photonics (2)

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2015).
[Crossref]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, Ph. Grangier, and E. Diamanti, “Experimental Demonstration of Long-Distance Continuous-Variable Quantum Key Distribution,” Nat. Photonics 7, 378 (2013).
[Crossref]

Nature (1)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref] [PubMed]

New J. Phys (1)

F. Xu, B. Qi, and H. K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys 12(11), 63–66 (2010).
[Crossref]

New J. Phys. (1)

M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012).
[Crossref]

Opt. Commun. (2)

S. K. Chong and T. H. wang, “Quantum key agreement protocol based on BB84,” Opt. Commun. 283(6), 1192–1195 (2010).
[Crossref]

J. Lin, H. Y. Tseng, and T. Hwang, “Intercept-resend attacks on Chen et al.’s quantum private comparison protocol and the improvements,” Opt. Commun. 284(9), 2412–2414 (2011).
[Crossref]

Opt. Express (1)

H.-C. Park, M. Lu, E. Bloch, T. Reed, Z. Griffith, L. Johansson, L. Coldren, and M. Rodwell, “40 Gbit/s Coherent Optical Receiver Using a Costas Loop,” Opt. Express 20, 197 (2012).
[Crossref]

Opt. Lett. (1)

Optical Technique (1)

M. A. Jing, Z. G. Yu, and Harbin, “Analysis of quantum bit error rate based on single-photon source with poisson distribution,” Optical Technique,  32, 101–104 (2006)

Phys Rev. A (1)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys Rev. A 87, 062313 (2013).
[Crossref]

Phys. Rev. A (5)

X. C. Ma, S. H. Sun, M. S. Jiang, and L. M. Liang, “Wavelength attack on practical continuous-variable quantum-key-distribution system with a heterodyne protocol,” Phys. Rev. A 87, 052309 (2013).
[Crossref]

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

L. Pezzè, “Sub-Heisenberg phase uncertainties,” Phys. Rev. A 88, 060101 (2013).
[Crossref]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A,  86, 22338 (2012).
[Crossref]

B. Xu, C. Tang, H. Chen, W. Zhang, and F. Zhu, “Improve the maximum transmission distance of four-state continuous variable quantum key distribution by using a noiseless linear amplifier,” Phys. Rev. A,  378, 2808–2812 (2012).

Phys. Rev. Lett. (7)

N. Wiebe and C. Granade, “Efficient bayesian phase estimation,” Phys. Rev. Lett. 117, 010503 (2015).
[Crossref]

V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref] [PubMed]

A. M. Lance, T. Symul, and V. Sharma, “No-switching quantum key distribution using broadband modulated coherent light,” Phys. Rev. Lett. 95(18), 180503 (2005).
[Crossref] [PubMed]

Raúl García-Patrón and Nicolas J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97 (19), 190503 (2006).
[Crossref] [PubMed]

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. Obrien, and M. G. Thompson, “Experimental bayesian quantum phase estimation on a silicon photonic chip,” Phys. Rev. Lett. 118, 100503 (2017).
[Crossref] [PubMed]

N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, “Hamiltonian learning and certication using quantum resources,” Phys. Rev. Lett. 112, 190501 (2014).
[Crossref]

Phys. Rev. X (2)

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5, 041009 (2015).

D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R.M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5, 041010 (2015).

Phys. Rev.A (1)

J. Z. Huang, C. Weedbrook, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, G. C. Guo, and Z. F. Han, “Quantum hacking on continuous-variable quantum key distribution system using a wavelength attack,” Phys. Rev.A 87, 062329 (2013).
[Crossref]

Physical Review A (1)

Y. Zhao, C. H. F. Fung, B. Qi, C. Chen, and H. Lo, “Quantum hacking : Experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Physical Review A 78(4). 042333 (2008).
[Crossref]

Publications of the American Statistical Association (1)

F. J. Samaniego, “Maximum likelihood estimation for binomially distributed signals in discrete noise,” Publications of the American Statistical Association 75, 117–121 (1980).
[Crossref]

Quantum Communications and Quantum Imaging IX (1)

F. Daneshgaran, M. T. Delgado, and M. Mondin, “Improved key rates for quantum key distribution employing soft metrics using Bayesian inference with photon counting detectors,” Quantum Communications and Quantum Imaging IX,  8163, 113–122 (2011).

Quantum Information & Computation (1)

K. M. Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quantum Information & Computation 14, 306 (2014).

Rev. Mod. Phys. (3)

L. B. Samuel and V. L. Peter, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. 81, 1301 (2009).
[Crossref]

Technometrics (1)

P. C. Consul and G. C. Jain, “A Generalization of the Poisson Distribution,” Technometrics 15, 791–799 (1973).
[Crossref]

Other (2)

S. Ren, R. Kumar, A. Wonfor, X. Tang, R. Penty, and I. White, “Reference pulse attack on continuous-variable quantum key distribution with local local oscillator,” (2017).

N. D Megill and M. Pavicic, “Estimating Bernoulli trial probability from a small sample,” https://arxiv.org/abs/1105.1486 (2011).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1 The constellation of the four-state CVQKD scheme.
Fig. 2
Fig. 2 The experiment step for the four-state CVQKD protocol. Alice sends coherent states through the noisy quantum channel, and then the measurement of phase shift should be performed with photon number resolving detector at the receiver. CW Laser, continuous-wave laser; AM, amplitude modulation; PM, phase modulation; PD, Photodiode; PNRD, photon number resolving detector.
Fig. 3
Fig. 3 The function of probability density function for phase drift. The detected photons at other three states are set as ni = 3 with i ∈ {1, 2, 3}, respectively.
Fig. 4
Fig. 4 The probability density function of phase shift at different detected photons n0. The detected photons are set to (a) n0 = 4, (b) n0 = 6, (c) n0 = 8 and (d) n0 = 10, respectively.
Fig. 5
Fig. 5 The mean quantum bit error rate (QBER) with four-state CVQKD protocol. Here, n represents the transmitted photons per pulse, and the detected photons per pulse are denoted as Nc. If the channel is lossless, two parameters are satisfied n = Nc. If the channel is loss, two parameters are satisfied n > Nc. Comprehensively considering both the lossless channel and lossy channel, two parameters are subject to the restriction nNc.
Fig. 6
Fig. 6 The mean quantum bit error rate (QBER). Here, Nc represents the detected photons per pulse, and the transmitted photons per pulse is set as n = 5.
Fig. 7
Fig. 7 Inference made with the posterior distribution of phase variance. The prior distribution satisfies (μ, σ2) ∼ ��(0, 103).
Fig. 8
Fig. 8 The different parameter estimation procedure. Here, the parameter estimation procedure is repeated with the different initial prior distribution, such as ��(μ, σ2) ∼ (0, 10−3), ��(μ, σ2) ∼ (0, 10−2), ��(μ, σ2) ∼ (0, 10−1), ��(μ, σ2) ∼ (0, 1), ��(μ, σ2) ∼ (0, 102) and ��(μ, σ2) ∼ (0, 103), respectively.
Fig. 9
Fig. 9 Inferences made with model prediction. The dataset (points) are utilized to estimate the posterior distribution of phase variance, which is analyzed in section 4.1. Subsequently, the posterior distribution of phase variance are utilized to generate the cumulative standard normal distribution of the model prediction.
Fig. 10
Fig. 10 The secret key rate for CVQKD protocol. Parameters are given as follows, α = 0.2 dB/km, vel = 0.1, = 0.05, η = 0.5, β = 0.95 and VA = 0.3.

Tables (3)

Tables Icon

Table 1 Bayesian phase estimation algorithm and Bayesian phase estimation theory

Tables Icon

Table 2 Algorithm 1. Bayesian phase estimation.

Tables Icon

Table 3 Algorithm 2. Bayesian prior distribution updating function

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

𝒫 ( ϕ | ) = 𝒫 ( | ϕ ) 𝒫 ( ϕ ) 𝒫 ( | ϕ ) 𝒫 ( ϕ ) d ϕ .
𝒫 ( | α 0 | ϕ ) = 1 4 ( 1 + e Δ 2 cos ( ϕ ) ) , 𝒫 ( | α 1 | ϕ ) = 1 4 ( 1 + e Δ 2 sin ( ϕ ) ) , 𝒫 ( | α 2 | ϕ ) = 1 4 ( 1 e Δ 2 cos ( ϕ ) ) , 𝒫 ( | α 3 | ϕ ) = 1 4 ( 1 e Δ 2 sin ( ϕ ) ) .
𝒫 B ( ϕ ; 𝒩 ) = i = 0 3 𝒫 ( | α i | ϕ ) n i ,
0 2 π 𝒫 B ( ϕ ; 𝒩 ) d ϕ = 1 .
Φ ( ) = 0 2 π ϕ 𝒫 ( ϕ | ) d ϕ .
( Δ 2 Φ ( ) ) ϕ | = 0 2 π ( ϕ Φ ( ) ) 2 𝒫 ( ϕ | ) d ϕ ,
( Δ 2 Φ ) , ϕ | ϕ = ( Δ 2 Φ ( ) ) ϕ | 𝒫 ( | ϕ ) = 0 2 π 𝒫 ( ϕ | ) 𝒫 ( | ϕ ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( ϕ | ) 𝒫 ( | ϕ ) = 𝒫 ( ϕ , | ϕ ) .
( Δ 2 Φ ) , ϕ , ϕ = 0 2 π ( Δ 2 Φ ) , ϕ | ϕ 𝒫 ( ϕ ) d ϕ = 0 2 π 0 2 π 𝒫 ( ϕ | ) 𝒫 ( | ϕ ) ( ϕ Φ ( ) ) 2 𝒫 ( ϕ ) d ϕ d ϕ = 0 2 π 𝒫 ( ϕ | ) 𝒫 ( ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( ) = 0 2 π 𝒫 ( | ϕ ) 𝒫 ( ϕ ) d ϕ .
( Δ 2 Φ ) , ϕ = 0 2 π 𝒫 ( , ϕ ) ( ϕ Φ ( ) ) 2 d ϕ ,
𝒫 ( , ϕ ) = 𝒫 ( ϕ | ) 𝒫 ( ) .
𝒫 ( | α 0 | ϕ 0 ) = 1 4 ( 1 + e Δ 2 cos ( π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 1 | ϕ 1 ) = 1 4 ( 1 + e Δ 2 sin ( 3 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 2 | ϕ 2 ) = 1 4 ( 1 + e Δ 2 cos ( 5 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) , 𝒫 ( | α 3 | ϕ 3 ) = 1 4 ( 1 + e Δ 2 sin ( 7 π / 4 ) ) = 1 4 ( 1 + 2 2 e Δ 2 ) .
𝒫 i i = 𝒫 ( | α i | ϕ i ) ,
j = 0 3 𝒫 ( | α i | ϕ j ) = 1 .
𝒫 i j = j = 0 3 𝒫 ( | α i | ϕ j ) ,
𝒫 i j = 1 𝒫 i i .
QBER = n 𝒬 n n ,
𝒬 n = { k = ( n + 1 ) / 2 n C n k 𝒫 i j k ( 1 𝒫 i j ) n k n is odd , k = ( n + 2 ) / 2 n C n k 𝒫 i j k ( 1 𝒫 i j ) n k + 1 2 C n n / 2 𝒫 i j n / 2 ( 1 𝒫 i j ) n / 2 n is even .
n = e N c N c n n ! , C n k = n ! k ! ( n k ) ! n ! = 1 × 2 × 3 × ( n 1 ) × n ,
Δ ρ = { Δ ρ 1 , Δ ρ 2 , Δ ρ c } .
Δ ρ = ρ s ρ n ,
ρ s = { ρ s 1 , ρ s 2 , , ρ s c } ,
ρ n = { ρ n 1 , ρ n 2 , , ρ n c } ,
PC = { PC 1 , PC 2 , , PC c } .
PC i = 𝒫 ( x s > x n ) = Φ ( Δ ρ i 2 σ 2 ) ,
k i Binomial ( PC i , T ) ,
K = β I ( x : y ) S ( y : E ) ,
I ( x : y ) = 1 2 log 2 V + χ tot 1 + χ tot ,
S ( y : E ) = G ( λ 1 1 2 ) + G ( λ 2 1 2 ) G ( λ 1 3 2 ) G ( λ 4 1 2 ) ,
λ 1 , 2 = 1 2 ( A ± A 2 4 B ,
A = V 2 + T 2 ( V + χ line 2 2 T Z 4 2 ) , B = ( T V 2 + T V χ line T Z 4 2 ) 2 .
λ 3 , 4 = 1 2 ( C ± C 2 4 D ,
C = A χ hom + V B + T ( V + χ line ) T ( V + χ tot ) , D = B V + B χ hom T ( V + χ tot ) .

Metrics