Abstract

We analyze the effect of phase fluctuations in an optical communication scheme based on collective detection of sequences of binary coherent state symbols using linear optics and photon counting. When the phase noise is absent, the scheme offers qualitatively improved nonlinear scaling of the spectral efficiency with the mean photon number in the low-power regime compared to individual detection. We show that this feature, providing a demonstration of superaddivitity of accessible information in classical communication over quantum channels, is preserved if random phases imprinted on transmitted symbols fluctuate around a reference fixed over the sequence length.

© 2016 Optical Society of America

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  1. J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
    [Crossref]
  2. K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014).
    [Crossref]
  3. S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” MIT Res. Lab. Electron. Quart. Prog. Rep. 111, 115 (1973).
  4. R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
    [Crossref] [PubMed]
  5. A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problems of Information Transmission 9, 177–183 (1973).
  6. M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
    [Crossref]
  7. S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
    [Crossref] [PubMed]
  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. M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
    [Crossref] [PubMed]
  10. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 2006).
  11. 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 (2013).
    [Crossref]
  12. 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]
  13. 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]
  14. C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
    [Crossref]
  15. F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
    [Crossref]

2015 (1)

2014 (2)

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]

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]

2013 (2)

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[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 (2013).
[Crossref]

2012 (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]

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

2011 (1)

S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[Crossref] [PubMed]

2007 (1)

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
[Crossref] [PubMed]

1998 (1)

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[Crossref]

1990 (1)

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

1973 (2)

S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” MIT Res. Lab. Electron. Quart. Prog. Rep. 111, 115 (1973).

A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problems of Information Transmission 9, 177–183 (1973).

Andersen, U. L.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Banaszek, K.

M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
[Crossref] [PubMed]

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]

Baumgartner, G.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

Becerra, F. E.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[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 (2013).
[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 (2013).
[Crossref]

Cook, R. L.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
[Crossref] [PubMed]

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]

Dolinar, S. J.

S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” MIT Res. Lab. Electron. Quart. Prog. Rep. 111, 115 (1973).

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]

Fan, J.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

Geremia, J. M.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
[Crossref] [PubMed]

Gnauck, A. H.

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

Goldhar, J.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

Guha, S.

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, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[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]

Hirota, O.

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[Crossref]

Holevo, A. S.

A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problems of Information Transmission 9, 177–183 (1973).

Izutsu, M.

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[Crossref]

Jarzyna, M.

M. Jarzyna, P. Kuszaj, and K. Banaszek, “Incoherent on-off keying with classical and non-classical light,” Opt. Express 23(3), 3170–3175 (2015).
[Crossref] [PubMed]

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]

Kahn, J. M.

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

Kasper, B. L.

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

Kato, K.

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[Crossref]

Kitayama, K.

K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014).
[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]

Korotky, S. K.

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

Kosloski, J. T.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

Kuszaj, P.

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]

Leuchs, G.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Marquardt, Ch.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Martin, P. J.

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
[Crossref] [PubMed]

Migdall, A.

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

Müller, C. R.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[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 (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 (2013).
[Crossref]

Sasaki, M.

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[Crossref]

Takeoka, M.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Thomas, J. A.

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

Usuga, M. A.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Veselka, J. J.

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[Crossref]

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]

Wittmann, C.

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[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]

IEEE Photon. Technol. Lett. (1)

J. M. Kahn, A. H. Gnauck, J. J. Veselka, S. K. Korotky, and B. L. Kasper, “4-Gb/s PSK homodyne transmission system using phase-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 2(4), 285 (1990).
[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. 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]

MIT Res. Lab. Electron. Quart. Prog. Rep. (1)

S. J. Dolinar, “An optimum receiver for the binary coherent state quantum channel,” MIT Res. Lab. Electron. Quart. Prog. Rep. 111, 115 (1973).

Nature (1)

R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement,” Nature 446(7137), 774–777 (2007).
[Crossref] [PubMed]

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]

F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination,” Nature Photon. 7(2), 147–152 (2013).
[Crossref]

New J. Phys. (1)

C. R. Müller, M. A. Usuga, C. Wittmann, M. Takeoka, Ch. Marquardt, U. L. Andersen, and G. Leuchs, “Quadrature phase shift keying coherent state discrimination via a hybrid receiver,” New J. Phys. 14, 083009 (2012)
[Crossref]

Opt. Express (1)

Phys. Rev. A (2)

M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58(1), 146–158 (1998).
[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 (2013).
[Crossref]

Phys. Rev. Lett. (1)

S. Guha, “Structured optical receivers to attain superadditive capacity and the Holevo limit,” Phys. Rev. Lett. 106(24), 240502 (2011).
[Crossref] [PubMed]

Problems of Information Transmission (1)

A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problems of Information Transmission 9, 177–183 (1973).

Other (2)

K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014).
[Crossref]

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

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Figures (2)

Fig. 1
Fig. 1

Sequences of BPSK symbols ± shown on the left are prepared as rows (or equivalently columns) of a symmetric Hadamard matrix. A linear circuit, described by a rescaled Hadamard matrix, transforms them into the PPM format visualized as tall solid red pulses localized in single output bins. When sequences are subject to phase noise, some of the input sequence energy becomes evenly distributed across all other bins, depicted with light red pulses. Assuming no dark counts, the sequence is either identified correctly with probability p by the position of the detector click (solid black arrows) or erased when no photocounts are generated (dashed blue arrows). In the presence of phase noise, the fraction of the sequence energy spread over all remaining PPM bins may produce a click in a given wrong bin with a probability q (dotted purple arrows). Events when clicks occur in two or more bins are treated as erasures.

Fig. 2
Fig. 2

(a) Shannon mutual information I given in Eq. (7) as a function of the sequence length L for n ¯ = 10 5 and several dephasings σ calculated for p and q estimated in Eqs. (5) and (6) (solid lines) and using Eq. (3) expanded up to n ¯ 2 (dashed lines). The case σ = 1.7 illustrates the breakdown of approximations used in the analysis. (b) The ratio I(σ)/I0 characterizing the effect of phase fluctuations on mutual information as a function of the average photon number n ¯. Information I(σ) is optimized over the sequence length L treated as a continuous parameter taking bounds given in Eq. (5) and (6) (solid lines), numerical expansions up to n ¯ 2 (dashed lines), and the closed approximate formula from Eq. (9) (dotted lines). The reference value I0 has been obtained by optimizing numerically over L the exact expression for mutual information in the noiseless case. Arrows indicate asymptotic values e σ 2. For σ = 0.3 dashed and solid lines overlap.

Equations (9)

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

p ( ϕ ) = exp [ ϕ 2 / ( 2 σ 2 ) ] / 2 π σ 2 .
μ k ( l ) = n ¯ L | j = 1 L e i ϕ j h k j h j l | 2 .
p k ( l ) = ( 1 e μ k ( l ) ) j k e μ j ( l ) = ( e μ k ( l ) 1 ) j e μ j ( l ) = e L n ¯ ( e μ k ( l ) 1 ) ,
μ k ( l ) = n ¯ [ 1 + ( L δ k l 1 ) e σ 2 ] .
p exp [ ( L 1 ) n ¯ ( 1 e σ 2 ) ] exp ( L n ¯ )
q 1 e μ k ( l ) μ k ( l ) = n ¯ ( 1 e σ 2 ) .
I = P L log 2 L 1 L [ p + ( L 1 ) q ] H ( p p + ( L 1 ) q ) q ( 1 1 L ) log 2 ( 1 1 L ) ,
p e σ 2 L n ¯ 1 2 ( 2 e σ 2 1 ) ( e σ 2 L n ¯ ) 2 .
I ( σ ) e σ 2 n ¯ Π ( ( 2 e σ 2 ) n ¯ ) n ¯ H ( e σ 2 ) ,

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