Abstract

We analyze the performance of on-off keying (OOK) and its restricted version pulse position modulation (PPM) over a lossy narrowband optical channel under the constraint of a low average photon number, when direct detection is used at the output. An analytical approximation for the maximum PPM transmission rate is derived, quantifying the effects of photon statistics on the communication efficiency in terms of the g(2) second-order intensity correlation function of the light source. Enhancement attainable through the use of sub-Poissonian light is discussed.

© 2015 Optical Society of America

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References

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  1. C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66(2), 481–537 (1994).
    [Crossref]
  2. J. H Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Topics Quantum Electron. 15(6), 1547–1569 (2009).
    [Crossref]
  3. V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
    [Crossref] [PubMed]
  4. V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
    [Crossref]
  5. S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
    [Crossref]
  6. A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
    [Crossref]
  7. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
    [Crossref]
  8. Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
    [Crossref]
  9. P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316 (1964).
    [Crossref]
  10. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [Crossref]
  11. W. Schmunk and et al., “Photon number statistics of NV centre emission,” Metrologia 49(2), S156–S160 (2012).
    [Crossref]
  12. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006).
  13. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
    [Crossref]
  14. K. Tsujino, D. Fukuda, G. Fujii, S. Inoue, M. Fujiwara, M. Takeoka, and M. Sasaki, “Sub-shot-noise-limit discrimination of on-off keyed coherent signals via a quantum receiver with a superconducting transition edge sensor,” Opt. Express 18(8), 8107–8114 (2010).
    [Crossref] [PubMed]
  15. J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
    [Crossref]
  16. S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
    [Crossref]
  17. M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
    [Crossref]
  18. M. Takeoka and S. Guha, “Capacity of optical communication in loss and noise with general quantum Gaussian receivers,” Phys. Rev. A 89(4), 042309 (2014).
    [Crossref]

2014 (5)

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
[Crossref]

M. Takeoka and S. Guha, “Capacity of optical communication in loss and noise with general quantum Gaussian receivers,” Phys. Rev. A 89(4), 042309 (2014).
[Crossref]

2013 (1)

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

2012 (2)

W. Schmunk and et al., “Photon number statistics of NV centre emission,” Metrologia 49(2), S156–S160 (2012).
[Crossref]

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

2011 (2)

A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
[Crossref]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
[Crossref]

2010 (1)

2009 (1)

J. H Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Topics Quantum Electron. 15(6), 1547–1569 (2009).
[Crossref]

2004 (1)

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

1996 (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

1994 (1)

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66(2), 481–537 (1994).
[Crossref]

1964 (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316 (1964).
[Crossref]

1963 (1)

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

Antonelli, C.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

Assalini, A.

Banaszek, K.

M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
[Crossref]

Castelli, F.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

Caves, C. M.

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66(2), 481–537 (1994).
[Crossref]

Cerf, N. J.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

Chen, J.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

Cialdi, S.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006).

Demkowicz-Dobrzanski, R.

M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
[Crossref]

Drummond, P. D.

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66(2), 481–537 (1994).
[Crossref]

Dutton, Z.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

Fujii, G.

Fujiwara, M.

Fukuda, D.

García-Patrón, R.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

Giovannetti, V.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Glauber, R. J.

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

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

Guha, S.

M. Takeoka and S. Guha, “Capacity of optical communication in loss and noise with general quantum Gaussian receivers,” Phys. Rev. A 89(4), 042309 (2014).
[Crossref]

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
[Crossref]

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Habif, J. L.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
[Crossref]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

Holevo, A. S.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

Inoue, S.

Jarzyna, M.

M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
[Crossref]

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316 (1964).
[Crossref]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316 (1964).
[Crossref]

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

Kochman, Y.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Lazarus, R.

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Maccone, L.

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Mecozzi, A.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

Olivares, S.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

Paris, M. G. A.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

Sasaki, M.

Schmunk, W.

W. Schmunk and et al., “Photon number statistics of NV centre emission,” Metrologia 49(2), S156–S160 (2012).
[Crossref]

Shapiro, J. H

J. H Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Topics Quantum Electron. 15(6), 1547–1569 (2009).
[Crossref]

Shapiro, J. H.

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Shtaif, M.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

Takeoka, M.

M. Takeoka and S. Guha, “Capacity of optical communication in loss and noise with general quantum Gaussian receivers,” Phys. Rev. A 89(4), 042309 (2014).
[Crossref]

A. Waseda, M. Sasaki, M. Takeoka, M. Fujiwara, M. Toyoshima, and A. Assalini, “Numerical evaluation of PPM for deep-space links,” J. Opt. Commun. Netw. 3(6), 514–521 (2011).
[Crossref]

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
[Crossref]

K. Tsujino, D. Fukuda, G. Fujii, S. Inoue, M. Fujiwara, M. Takeoka, and M. Sasaki, “Sub-shot-noise-limit discrimination of on-off keyed coherent signals via a quantum receiver with a superconducting transition edge sensor,” Opt. Express 18(8), 8107–8114 (2010).
[Crossref] [PubMed]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006).

Toyoshima, M.

Tsujino, K.

Wang, L.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Waseda, A.

Winzer, P. J.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

Wornell, G. W.

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

Yuen, H. P.

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W Function,” Adv. Comput. Math. 5(4), 329–359 (1996).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

J. H Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Topics Quantum Electron. 15(6), 1547–1569 (2009).
[Crossref]

IEEE Trans. Inf. Theory (1)

Y. Kochman, L. Wang, and G. W. Wornell, “Toward photon-efficient key distribution over optical channels,” IEEE Trans. Inf. Theory 60(8), 4958–4972 (2014).
[Crossref]

J. Lightw. Technol. (1)

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Quantum limits on the energy consumption of optical transmission systems,” J. Lightw. Technol. 32(10), 1853–1860 (2014).
[Crossref]

J. Mod. Opt. (1)

S. Guha, J. L. Habif, and M. Takeoka, “Approaching Helstrom limits to optical pulse-position demodulation using single photon detection and optical feedback,” J. Mod. Opt. 58(3–4), 257–265 (2011).
[Crossref]

J. Opt. Commun. Netw. (1)

J. Phys. A : Math. Theor. (1)

M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrzański, “Dephasing in coherent communication with weak signal states,” J. Phys. A : Math. Theor. 47(27), 275302 (2014).
[Crossref]

Metrologia (1)

W. Schmunk and et al., “Photon number statistics of NV centre emission,” Metrologia 49(2), S156–S160 (2012).
[Crossref]

Nature Photon. (2)

J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver,” Nature Photon. 6(6), 374–379 (2012).
[Crossref]

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture,” Nature Photon. 8(10), 796–800 (2014).
[Crossref]

Opt. Express (1)

Phys. Rev. (2)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136(2A), A316 (1964).
[Crossref]

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

Phys. Rev. A (2)

M. Takeoka and S. Guha, “Capacity of optical communication in loss and noise with general quantum Gaussian receivers,” Phys. Rev. A 89(4), 042309 (2014).
[Crossref]

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87(5), 050303(R) (2013).
[Crossref]

Phys. Rev. Lett. (1)

V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, “Classical capacity of the lossy bosonic channel: the exact solution,” Phys. Rev. Lett. 92(2), 027902 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66(2), 481–537 (1994).
[Crossref]

Other (1)

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006).

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Figures (2)

Fig. 1
Fig. 1 (a) OOK with direct detection represented as a binary asymmetric channel. For the description of symbols, see the main text. (b) Left scale: the approximate analytical expression for photon information efficiency Π derived in Eq. (5) as a function of the average output photon number ηn̄ (red solid line) compared with the result of numerical optimization assuming Poissonian photon statistics for PPM (black solid line) and OOK (blue solid line) schemes. The blue dotted line depicts optimized PIE for OOK with Poisonian statistics and dark count probability 0.25ηn̄ per time bin. The grey region represents values of PIE beyond the capacity limit of a single-mode bosonic channel, equal to log2(1/ηn̄) + (1 + 1/ηn̄)log2(1 + ηn̄). Right scale: the optimal value of 1/p maximizing mutual information given by the approximate expression in Eq. (2) (red dashed line) compared to numerical results for PPM (black dashed line) and OOK (red dashed line) with Poissonian photon statistics.
Fig. 2
Fig. 2 Ratio of mutual information optimized over non-classical two-component mixtures of adjacent Fock states to that attainable with Poissonian statistics of the non-zero pulse for (a) PPM and (b) OOK schemes. The dashed lines indicate regions where single-photon states with μ = 1 are optimal in the non-classical case.

Equations (7)

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

ε 1 η n ^ + 1 2 η 2 : n ^ 2 : = 1 η μ + 1 2 g ( 2 ) η 2 μ 2 .
I PPM = η n ¯ ( 1 1 2 g ( 2 ) η μ ) log 2 μ n ¯ .
μ clas = 2 η g ( 2 ) [ W ( 2 e g ( 2 ) η n ¯ ) ] 1 .
I PPM clas = η n ¯ Π ( g ( 2 ) η n ¯ )
Π ( η n ¯ ) = { [ W ( 2 e η n ¯ ) ] 1 1 } log 2 [ η n ¯ 2 W ( 2 e η n ¯ ) ] .
μ 1 1 g ( 2 ) .
I PPM opt = { η n ¯ log 2 1 n ¯ if η 2 / ln 1 n ¯ , η n ¯ ( 1 + η 2 ) Π ( η n ¯ 1 + η / 2 ) if η < 2 / ln 1 n ¯ .

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