Abstract

Optical measurements in quantum communication and quantum radiometry are quite often based on two-photon correlated channels and the coincidence between them. To quantify the noise level in these measurements, the authors focus on the statistics of photon coincidence in a basic experimental setup, accounting for all experimental effects contributing to the noise increment. By the proposed theory the measurement performance is evaluated in terms of noise fluctuations. Particularly, an ultimate limit in noise reduction is established by the optimal estimation model herein presented.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
    [CrossRef]
  4. J. G. Rarity and P. R. Tapster, “Quantum communication,” Appl. Phys. B 55, 298–303 (1992).
    [CrossRef]
  5. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
    [CrossRef]
  6. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988).
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    [CrossRef] [PubMed]
  8. A. N. Penin and A. V. Sergienko, “Absolute standardless calibration of photodetectors based on quantum two-photon field,” Appl. Opt. 30, 3582–3588 (1991).
    [CrossRef] [PubMed]
  9. P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. H. Eberhard, and M. D. Petroff, “Absolute efficiency and time-response measurement of single-photon detectors,” Appl. Opt. 33, 1844–1853 (1994).
    [CrossRef] [PubMed]
  10. A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
    [CrossRef]
  11. S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
    [CrossRef]
  12. G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
    [CrossRef]
  13. G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
    [CrossRef]
  14. A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
    [CrossRef] [PubMed]
  15. A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
    [CrossRef]
  16. T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
    [CrossRef] [PubMed]
  17. D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
    [CrossRef] [PubMed]
  18. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
    [CrossRef] [PubMed]
  19. P. R. Tapster, J. G. Rarity, and P. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
    [CrossRef] [PubMed]
  20. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
    [CrossRef]
  21. G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
    [CrossRef]
  22. N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
    [CrossRef]
  23. For detailed discussion of this argument we refer to Subsection 3.C.
  24. M. M. Hayat, A. Joobeur, and B. E. A. Saleh, “Reduction of quantum noise in transmittance estimation using photon-correlated beams,” J. Opt. Soc. Am. A 16, 348–358 (1999).
    [CrossRef]
  25. M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
    [CrossRef]
  26. A. Migdall, National Institute of Standards and Technology, Gaithersburg, Maryland, and S. Castelletto, I. P. Degiovanni, and M. L. Rastello, Istituto Elettrotecnico Nazionale G. Ferraris, Torino, Italy (personal communication).
  27. S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
    [CrossRef]
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  29. S. Castelletto, I. P. Degiovanni, and M. L. Rastello, Istituto Elettrotecnico Nazionale G. Ferraris, “Statistical models for quantum cryptography experiments based on polarization entangled photons,” Tech. Rep. 644 (Marzo, Italy, 2002).
  30. J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science (Wiley, New York, 1977).
  31. To compare forwardly the result of εR (ηs πs ︿NP) with εR (ηs πs ︿NP)Gauss, we do not consider τs as an input quantity. To calculate the uncertainty in the case of a real measurement, it is obvious that τs should enter in the vector of input quantities.
  32. Incidentally, since in the two-photon beam setup the probe and the reference are separate channels, measurements on them can be performed at the same time, thus exploiting quantum correlation between measured quantities.
  33. To this respect, we point out that considerations regarding the reference as a metrological standard, i.e., accuracy and traceability, are addressed neither in the two-photon-beam nor in the single-beam configuration. Note that in the two-photon-beam arrangement the measurement is standardless, whereas in the single-beam setup a necessary further traceability uncertainty should be added to the statistical uncertainty considered here.
  34. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, New York, 1984), pp. 175–179.
  35. A. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [CrossRef] [PubMed]
  36. G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
    [CrossRef] [PubMed]
  37. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
    [CrossRef] [PubMed]

2000 (7)

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
[CrossRef]

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

1999 (4)

M. M. Hayat, A. Joobeur, and B. E. A. Saleh, “Reduction of quantum noise in transmittance estimation using photon-correlated beams,” J. Opt. Soc. Am. A 16, 348–358 (1999).
[CrossRef]

M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
[CrossRef]

G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
[CrossRef]

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

1998 (2)

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

1996 (2)

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

1994 (2)

1992 (2)

J. G. Rarity and P. R. Tapster, “Quantum communication,” Appl. Phys. B 55, 298–303 (1992).
[CrossRef]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

1991 (2)

1987 (2)

J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987).
[CrossRef] [PubMed]

1986 (1)

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmissions measurements,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

1984 (1)

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Atature, M.

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

Berglund, A.

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

Boeuf, N.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Branning, D.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Brassard, G.

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

Brida, G.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
[CrossRef]

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Castelletto, S.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
[CrossRef]

G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
[CrossRef]

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

Chaperot, I.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Chiao, R. Y.

Datla, R. U.

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

Dauler, E.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Degiovanni, I. P.

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
[CrossRef]

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

Eberhard, P. H.

Ekert, A.

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

Ekert, A. K.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

Godone, A.

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

Guerin, S.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Hayat, M. M.

M. M. Hayat, A. Joobeur, and B. E. A. Saleh, “Reduction of quantum noise in transmittance estimation using photon-correlated beams,” J. Opt. Soc. Am. A 16, 348–358 (1999).
[CrossRef]

M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
[CrossRef]

Jaeger, G.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

Jakeman, E.

J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmissions measurements,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

Jennewein, T.

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

Joobeur, A.

Kwiat, P.

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. H. Eberhard, and M. D. Petroff, “Absolute efficiency and time-response measurement of single-photon detectors,” Appl. Opt. 33, 1844–1853 (1994).
[CrossRef] [PubMed]

Lutkenhaus, N.

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

Mandel, L.

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Migdall, A.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Migdall, A. L.

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

Mor, T.

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

Muller, A.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Naik, D.

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

Novero, C.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
[CrossRef]

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

Orszak, J. S.

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

Owens, P.

P. R. Tapster, J. G. Rarity, and P. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

Palma, G. M.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

Pedrotti, L. M.

M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
[CrossRef]

Penin, A. N.

Peterson, C.

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

Petroff, M. D.

Rarity, J. G.

P. R. Tapster, J. G. Rarity, and P. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Quantum communication,” Appl. Phys. B 55, 298–303 (1992).
[CrossRef]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987).
[CrossRef] [PubMed]

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmissions measurements,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

Rastello, M. L.

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
[CrossRef]

G. Brida, S. Castelletto, C. Novero, and M. L. Rastello, “Quantum efficiency measurement of photodetectors by means of correlated photons,” J. Opt. Soc. Am. B 16, 1623–1627 (1999).
[CrossRef]

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

Ridley, K. D.

Saleh, B. E. A.

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

M. M. Hayat, A. Joobeur, and B. E. A. Saleh, “Reduction of quantum noise in transmittance estimation using photon-correlated beams,” J. Opt. Soc. Am. A 16, 348–358 (1999).
[CrossRef]

Sanders, B.

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

Sergienko, A. V.

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

A. N. Penin and A. V. Sergienko, “Absolute standardless calibration of photodetectors based on quantum two-photon field,” Appl. Opt. 30, 3582–3588 (1991).
[CrossRef] [PubMed]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Shih, Y. H.

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

Simon, C.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

Simon, G.

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

Steinberg, A. M.

Tapster, P. R.

P. R. Tapster, J. G. Rarity, and P. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Quantum communication,” Appl. Phys. B 55, 298–303 (1992).
[CrossRef]

J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987).
[CrossRef] [PubMed]

Teich, M. C.

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

Torres, S. N.

M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
[CrossRef]

Walton, Z.

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

Weihs, G.

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Weinfurther, H.

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

White, A.

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

Zeilinger, A.

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. B (1)

J. G. Rarity and P. R. Tapster, “Quantum communication,” Appl. Phys. B 55, 298–303 (1992).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Metrologia (4)

G. Brida, S. Castelletto, I. P. Degiovanni, C. Novero, and M. L. Rastello, “Quantum efficiency and dead time measurement of single-photon photodiodes: a comparison between two techniques,” Metrologia 37, 625–628 (2000).
[CrossRef]

A. L. Migdall, R. U. Datla, A. V. Sergienko, J. S. Orszak, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1996).
[CrossRef]

S. Castelletto, A. Godone, C. Novero, and M. L. Rastello, “Biphoton fields for quantum-efficiency measurement,” Metrologia 32, 501–503 (1996).
[CrossRef]

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, “Theoretical aspects of photon number measurement,” Metrologia 37, 613–616 (2000).
[CrossRef]

Opt. Commun. (3)

M. M. Hayat, S. N. Torres, and L. M. Pedrotti, “Theory of photon coincidence statistics in photon-correlated beams,” Opt. Commun. 169, 275–287 (1999).
[CrossRef]

E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmissions measurements,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

J. G. Rarity, P. R. Tapster, and E. Jakeman, “Observation of sub-Poissonian light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

Opt. Eng. (1)

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guerin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of non-collinear phase-matching in uniaxial and biaxial crystals,” Opt. Eng. 39, 1016–1024 (2000).
[CrossRef]

Phys. Rev. A (1)

A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625 (1999).
[CrossRef]

Phys. Rev. Lett. (11)

T. Jennewein, G. Simon, G. Weihs, H. Weinfurther, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[CrossRef] [PubMed]

D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, “Entangled state quantum cryptography: eavesdropping, on the Ekert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[CrossRef] [PubMed]

P. R. Tapster, J. G. Rarity, and P. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998).
[CrossRef]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurther, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[CrossRef]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
[CrossRef] [PubMed]

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

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

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations of practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Other (10)

A. Migdall, National Institute of Standards and Technology, Gaithersburg, Maryland, and S. Castelletto, I. P. Degiovanni, and M. L. Rastello, Istituto Elettrotecnico Nazionale G. Ferraris, Torino, Italy (personal communication).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

S. Castelletto, I. P. Degiovanni, and M. L. Rastello, Istituto Elettrotecnico Nazionale G. Ferraris, “Statistical models for quantum cryptography experiments based on polarization entangled photons,” Tech. Rep. 644 (Marzo, Italy, 2002).

J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science (Wiley, New York, 1977).

To compare forwardly the result of εR (ηs πs ︿NP) with εR (ηs πs ︿NP)Gauss, we do not consider τs as an input quantity. To calculate the uncertainty in the case of a real measurement, it is obvious that τs should enter in the vector of input quantities.

Incidentally, since in the two-photon beam setup the probe and the reference are separate channels, measurements on them can be performed at the same time, thus exploiting quantum correlation between measured quantities.

To this respect, we point out that considerations regarding the reference as a metrological standard, i.e., accuracy and traceability, are addressed neither in the two-photon-beam nor in the single-beam configuration. Note that in the two-photon-beam arrangement the measurement is standardless, whereas in the single-beam setup a necessary further traceability uncertainty should be added to the statistical uncertainty considered here.

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

D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988).

For detailed discussion of this argument we refer to Subsection 3.C.

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Figures (12)

Fig. 1
Fig. 1

Typical experimental scheme for optical measurement with two-photon correlated beams.

Fig. 2
Fig. 2

Relative noise fluctuation, R(ηsπsˆML), of maximum-likelihood best estimators ηsπsˆML, is plotted versus ρi and ρs/ρi with ηs=0.6, τs=τi=0.6, αi=0.995, ηi=0.5, Di=1600 ns, Ds=30 ns, and w=4 ns, for nonextending dead-time case.

Fig. 3
Fig. 3

Relative noise fluctuation R(ηsπsˆML) is plotted versus Di for three different ratios ρs/ρi=2, 10, 20 for both extending and nonextending dead-time cases with ηs=0.8, τs=1, ρi=250 kHz, and Ds=30 ns. The other parameters’ fixed values are αi=0.999, τi=1, ηi=0.8, and w=4 ns.

Fig. 4
Fig. 4

Relative noise fluctuation R(ηsπsˆML) is plotted versus ρi and αi for the nonextending dead-time case. The other parameters’ fixed values are ρs/ρi=1.5, ηs=ηi=0.6, Di=Ds=30 ns, τs=τi=1, and w=1 ns.

Fig. 5
Fig. 5

Relative discrepancy between estimates ΔR(ηsπsˆ) (straight line) and associated relative uncertainty R(ηsπsˆML) (dashed curve) and R(ηsπsˆNP)Gauss (dotted curve) are plotted versus ρi for ρs/ρi=1.5 in the nonextending dead-time case. The other parameters have fixed values ηs=ηi=0.6, τs=τi=1, αi=0.999, Di=Ds=30 ns, and w=1 ns.

Fig. 6
Fig. 6

Relative uncertainties R(ηsπsˆML) (straight line), R(ηsπsˆNP)Gauss (dashed curve) R(ηsπsˆNP)Gaussuncorr (dotted curve) are plotted versus associated estimates ηsˆ for ρs/ρi=1.5 in the nonextending dead-time case. The other parameters have fixed values ηi=0.6, τs=τi=1, αi=0.999, Di=1600 ns, Ds=30 ns, w=4 ns, ρi=250 kHz.

Fig. 7
Fig. 7

Relative uncertainties R(ηsπsˆML) in the following conditions: ρs/ρi=1, ηi=0.6, αi=1, Di=30 ns, and w=1 ns (straight line), ρs/ρi=10, ηi=0.6, αi=0.96, Di=1600 ns, w=4 ns [long-dashed curve (b)], ρs/ρi=10, ηi=0.4, αi=0.96, Di=1600 ns, w=4 ns [short-dashed curve (c)], together with R(ηlπlˆ)total for ηl=ηs=0.6, τl=τs=1, Dl=Ds=30 ns (dotted curve), are plotted versus the reference/trigger counts.

Fig. 8
Fig. 8

Relative noise fluctuation, R(τsˆML), of maximum-likelihood best estimator τsˆML is plotted versus ρi and ρs/ρi with ηs=0.6, τs=τi=0.6, αi=0.995, ηi=0.5, Di=1600 ns, Ds=30 ns, and w=4 ns for extending (a) and nonextending (b) dead-time cases.

Fig. 9
Fig. 9

Relative noise fluctuation, R(τsˆML), is plotted versus Ds and Di for two different ratios ρs/ρi=2 (a) and 15 (b) for extending the dead-time case with ηs=τs=0.8 and ρi=400 kHz. The other parameters’ fixed values are αi=0.999, τi=1, ηi=0.8, and w=4 ns.

Fig. 10
Fig. 10

Relative discrepancy between estimates ΔR(τsˆ) (straight line) and associated relative uncertainty R(τsˆML) (short-dashed curve), R(τsˆML)total (dashed curve), and R(τsˆNP)Gauss (dotted curve) are plotted versus ρi for ρs/ρi=1.5 in the nonextending dead-time case. The other parameters have fixed values ηs=ηi=0.8, τs=0.5, τi=1, αi=0.999, Di=Ds=30 ns, and w=1 ns.

Fig. 11
Fig. 11

QBER(A) is plotted versus ρi in the case of ρs/ρi=1, 1.5 and αi=0.999,0.5. The other parameters have fixed values ηs=ηi=0.8, τi=1, Di=1600 ns, Ds=30 ns, and w=4 ns.

Fig. 12
Fig. 12

QBER(A) [(B)] is plotted versus ρi in the condition (a), ρs/ρi=1, ηs=ηi=1, Di=Ds=w=1 ns dotted line [straight line], and in the realistic ones (b), ρs/ρi=1.5, ηs=ηi=0.8, τi=1, αi=0.999, Di=1600 ns, Ds=30 ns, and w=4 ns long-dashed line [short-dashed line], respectively. The other parameters have fixed values αi=τi=1.

Tables (2)

Tables Icon

Table 1 Calculation of R(ηsπsˆML) and ηsπsˆML from Experimental Data Obtained in the Calibration of DUT A and DUT B

Tables Icon

Table 2 Calculation of R(ηsπsˆML), R(ηsπsˆNP) and R(ηsπsˆNP)Gauss from Experimental Data Obtained in the Calibration of DUT A and DUT B

Equations (35)

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ρx=αxρx+(1-αx)ρx,
Px(mα)=nα=mαP(nα)B(nα, mα, τxηx)=(αxρxηxτxT)mαexp(-αxρxηxτxT)/mα!,
P(mc)=mα=mcPi(mα)B(mα, mc, τsηs)=(τsηsαiρiτiηiT)mcmc!exp(-τsηsαiρiηiτiT).
Pxext(mα)=mα=mαPx(mα)B(mα, mα, πxext)=Px(mα),
Pxnext(mα)=Px(mα, αxρxηxτxT, Dx)=γ[mα;αxρxηxτx(T-mαDx)]/(mα-1)!-γ{mα+1;αxρxηxτx×[T-(mα+1)Dx]}/mα!
P(c)(mc)mα=mcPi(mα)B(mα, mc, τsηsπs*),
ρR,i=ρiηiτi(1-αiηsτsπs*),
ρR,s=ρsηsτs(1-αsηiτiπi*).
Pi(mR, ρR,iT, Di)mR=mRP(mR)B(mR, mR, πi*).
ps=mR=1Ps(mR, ρR,sw, Ds),
Pi(a)(ma)=mR=maPi(mR, ρR,iT, Di)B(mR, ma, ps),
Pi(mc)=ma=0mc=0δmc, ma+mcPi(a)(ma)P(c)(mc).
P(mc)y=0.
R(yˆML)=yˆML2-yˆML2yˆML21/2,
ρc,iT-mc=0.
ηsπsˆMLmcτsαiρiηiπi*τiT[1+ρiw(αs-1-ηiτiπi*)].
R(ηsπsˆML)=mc-1/2{Tηsπsˆρiηiπi*τiτsαi×[1+wρi(ηiτiπi*-αs-1)×(αiτsηsπsˆ-1)]}-1/2.
ηsπsˆ=1τsmcmi.
ηsπsˆ=1τsmc-Ami-mN,i,
A=(mi-mc)mswT+12mswT2,
ηsπsˆNPmcτsαiρiηiπi*τiT{1+ρiw[αs-1-mc(ηiπi*τiρiαsT)-1]}=ηsπsˆML1+ρiw(αs-1-ηiτiπi*)1+ρiw[αs-1-mc(ηiπi*τiρiαsT)-1].
R(ηsπsˆNP)=R(ηsπsˆML)×1-mcρiηiπi*τiT[1+αs/(ρiw)]-1>R(ηsπsˆML),
R(ηsπsˆNP)Gauss=1ηsπsˆNP(J·C·JT)1/2,
C=mcmcmcmsmN,iw/TmimcmN,imsmsmN,iw/TmN,i.
Pl(ml)=nl=mlP(nl, ρl)B(nl, ml, τlηlπl).
R(ηlπlˆ)total=1+ηlηlπlτlηlπlρlT1/2.
ρiηiτiπi*{1-(1-αiηsτsπs*)×exp[-ρsηsτsπs*(1-αsηiτiπi*)w]}T-mc=0.
τsˆNP=mcon-Aonmcoff-Aoff1-msoffDs/T1-msonDs/T,
C=Con00Coff,
Cx=mcxmcxmcxmixmcxmsx,
R(βˆML)=1-QBERK×QBER1/2.
ρC=14αiρiηiτiπi*ηsτsπs*,
ρA=14ρiηiτiπi*ps+14ρiηiτiπi*ps(1-αiηsτsπs*),
QBER(A)=mA,mc=ps2ps-psαiηsτsπs*+αiηsτsπs*,
QBER(B)=mA,+mi,-mc,mi=ps+1-αiηsτsπs*2,

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