Abstract

The noise properties of phase-insensitive and phase-sensitive optical transmission links are described in detail, for Gaussian input signals. Formulas are derived for the quadrature covariance matrices of the output signals, which allow one to quantify the noise figures of the links and the fidelities of transmission. Another formula is derived, which relates the density operator of an output signal, in the number-state representation, to its covariance matrix. This density matrix allows one to quantify the decrease in coherence and changes in photon-number probabilities associated with transmission. Based on the aforementioned performance metrics, links with distributed phase-sensitive amplification perform significantly better than other links.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).
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  5. R. M. Gagliardi and S. Karp, Optical Communications, 2nd Ed. (Wiley, 1995).
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  7. C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
    [Crossref]
  8. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [Crossref]
  9. C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
    [Crossref]
  10. F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
    [Crossref]
  11. D. C. Burnham and D. L. Weinberg, “Obervation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
    [Crossref]
  12. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon Technol. Lett. 14, 983–985 (2002).
    [Crossref]
  13. H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd Ed. (Wiley, 2004).
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    [Crossref]
  16. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd Ed. (Wiley, 2006).
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    [Crossref]
  18. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [Crossref]
  19. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
    [Crossref]
  20. R. E. Slusher and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
    [Crossref]
  21. C. R. Giles and E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
    [Crossref]
  22. H. Yuen, “Reduction of quantum fluctuations and suppression of the Gordon–Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. 12, 73–75 (1992).
    [Crossref]
  23. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
    [Crossref] [PubMed]
  24. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
    [Crossref] [PubMed]
  25. C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006).
    [Crossref]
  26. M. Vasilyev and N. Stelmakh, “Squeezing and fiber-optic communications,” Int J. Mod. Phys. B 20, 1536–1542 (2006).
    [Crossref]
  27. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
    [Crossref] [PubMed]
  28. C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011).
    [Crossref] [PubMed]
  29. A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd Ed. (McGraw-Hill, 1991).
  30. R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
    [Crossref]
  31. H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,” J. Phys. A: Math. Gen. 31, 3659–3663 (1998).
    [Crossref]
  32. C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” IEEE J. Sel. Top. Quantum Electron. 18, 794–811 (2012).
    [Crossref]
  33. N. N. Lebedev, Special Functions and their Applications (Dover, 1972).
  34. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
    [Crossref]
  35. A. Yariv, “Signal-to-noise considerations in fiber links with periodic or distributed optical amplification,” Opt. Lett. 15, 1064–1066 (1990).
    [Crossref] [PubMed]
  36. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
    [Crossref]
  37. F. Caruso, V. Giovannetti, and A. S. Kholevo, “One-mode bosonic Gaussian channels: a full weak-degradability classification,” New J. Phys 8, 310 (2006).
    [Crossref]
  38. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [Crossref]
  39. J. Paye, “The chroncyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
    [Crossref]
  40. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007).
  41. K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
    [Crossref]
  42. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
    [Crossref]
  43. B. M. Garraway and P. L. Knight, “Quantum phase distributions and quasidistributions,” Phys. Rev. A 46, R5346–R5349 (1992).
    [Crossref] [PubMed]
  44. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [Crossref] [PubMed]
  45. E. Arthurs and J. L. Kelley, “On the simultaneous measurement of a pair of conjugate variables,” Bell Sys. Tech. J. 44, 725–729 (1965).
    [Crossref]

2014 (1)

F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
[Crossref]

2012 (2)

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” IEEE J. Sel. Top. Quantum Electron. 18, 794–811 (2012).
[Crossref]

2011 (1)

2010 (1)

2006 (3)

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006).
[Crossref]

M. Vasilyev and N. Stelmakh, “Squeezing and fiber-optic communications,” Int J. Mod. Phys. B 20, 1536–1542 (2006).
[Crossref]

F. Caruso, V. Giovannetti, and A. S. Kholevo, “One-mode bosonic Gaussian channels: a full weak-degradability classification,” New J. Phys 8, 310 (2006).
[Crossref]

2005 (3)

2002 (1)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon Technol. Lett. 14, 983–985 (2002).
[Crossref]

1998 (1)

H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,” J. Phys. A: Math. Gen. 31, 3659–3663 (1998).
[Crossref]

1994 (2)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[Crossref]

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

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

1992 (3)

B. M. Garraway and P. L. Knight, “Quantum phase distributions and quasidistributions,” Phys. Rev. A 46, R5346–R5349 (1992).
[Crossref] [PubMed]

J. Paye, “The chroncyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[Crossref]

H. Yuen, “Reduction of quantum fluctuations and suppression of the Gordon–Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. 12, 73–75 (1992).
[Crossref]

1991 (2)

C. R. Giles and E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[Crossref]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[Crossref]

1990 (2)

A. Yariv, “Signal-to-noise considerations in fiber links with periodic or distributed optical amplification,” Opt. Lett. 15, 1064–1066 (1990).
[Crossref] [PubMed]

R. E. Slusher and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[Crossref]

1985 (1)

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
[Crossref]

1979 (1)

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[Crossref]

1970 (1)

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

1969 (2)

K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[Crossref]

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

1965 (1)

E. Arthurs and J. L. Kelley, “On the simultaneous measurement of a pair of conjugate variables,” Bell Sys. Tech. J. 44, 725–729 (1965).
[Crossref]

1963 (1)

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

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell. Sys. Tech. J. 28, 379–423 and 623–656 (1948).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 4th Ed. (Wiley, 2010).
[Crossref]

Alic, N.

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” IEEE J. Sel. Top. Quantum Electron. 18, 794–811 (2012).
[Crossref]

C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011).
[Crossref] [PubMed]

Arthurs, E.

E. Arthurs and J. L. Kelley, “On the simultaneous measurement of a pair of conjugate variables,” Bell Sys. Tech. J. 44, 725–729 (1965).
[Crossref]

Bachor, H. A.

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd Ed. (Wiley, 2004).

Bastiaans, M. J.

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Burnham, D. C.

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

Cahill, K. E.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[Crossref]

Caruso, F.

F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
[Crossref]

F. Caruso, V. Giovannetti, and A. S. Kholevo, “One-mode bosonic Gaussian channels: a full weak-degradability classification,” New J. Phys 8, 310 (2006).
[Crossref]

Caves, C. M.

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

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Chraplyvy, A. R.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd Ed. (Wiley, 2006).

Desurvire, E.

C. R. Giles and E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[Crossref]

Drummond, P. D.

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

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon Technol. Lett. 14, 983–985 (2002).
[Crossref]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications, 2nd Ed. (Wiley, 1995).

Garcia-Patron, R.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Garraway, B. M.

B. M. Garraway and P. L. Knight, “Quantum phase distributions and quasidistributions,” Phys. Rev. A 46, R5346–R5349 (1992).
[Crossref] [PubMed]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Giles, C. R.

C. R. Giles and E. Desurvire, “Propagation of signal and noise in concatenated erbium-doped fiber optical amplifiers,” J. Lightwave Technol. 9, 147–154 (1991).
[Crossref]

Giovannetti, V.

F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
[Crossref]

F. Caruso, V. Giovannetti, and A. S. Kholevo, “One-mode bosonic Gaussian channels: a full weak-degradability classification,” New J. Phys 8, 310 (2006).
[Crossref]

Glauber, R. J.

K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[Crossref]

K. E. Cahill and R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969).
[Crossref]

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

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[Crossref]

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007).

Jopson, R. M.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006).
[Crossref]

Jozsa, R.

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[Crossref]

Karlsson, M.

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications, 2nd Ed. (Wiley, 1995).

Kelley, J. L.

E. Arthurs and J. L. Kelley, “On the simultaneous measurement of a pair of conjugate variables,” Bell Sys. Tech. J. 44, 725–729 (1965).
[Crossref]

Kholevo, A. S.

F. Caruso, V. Giovannetti, and A. S. Kholevo, “One-mode bosonic Gaussian channels: a full weak-degradability classification,” New J. Phys 8, 310 (2006).
[Crossref]

Knight, P. L.

B. M. Garraway and P. L. Knight, “Quantum phase distributions and quasidistributions,” Phys. Rev. A 46, R5346–R5349 (1992).
[Crossref] [PubMed]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Kumar, P.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon Technol. Lett. 14, 983–985 (2002).
[Crossref]

Lebedev, N. N.

N. N. Lebedev, Special Functions and their Applications (Dover, 1972).

Leonhardt, U.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

Lloyd, S.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Loudon, R.

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
[Crossref]

Lupo, C.

F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
[Crossref]

Mancini, S.

F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, “Quantum channels and memory effects,” Rev. Mod. Phys. 86, 1203–1259 (2014).
[Crossref]

Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
[Crossref]

McKinstrie, C. J.

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd Ed. (McGraw-Hill, 1991).

Paye, J.

J. Paye, “The chroncyclic representation of ultrashort light pulses,” IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[Crossref]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Radic, S.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259, 309–320 (2006).
[Crossref]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd Ed. (Wiley, 2004).

Raymer, M. G.

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[Crossref] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007).

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley, 2001), pp. 104–106.

Scutaru, H.

H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,” J. Phys. A: Math. Gen. 31, 3659–3663 (1998).
[Crossref]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell. Sys. Tech. J. 28, 379–423 and 623–656 (1948).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Sharping, J. E.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon Technol. Lett. 14, 983–985 (2002).
[Crossref]

Shih, Y.

Y. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC Press, 2011).

Slusher, R. E.

R. E. Slusher and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[Crossref]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[Crossref] [PubMed]

Stelmakh, N.

M. Vasilyev and N. Stelmakh, “Squeezing and fiber-optic communications,” Int J. Mod. Phys. B 20, 1536–1542 (2006).
[Crossref]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd Ed. (Wiley, 2006).

Tong, Z.

van Loock, P.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Vasilyev, M.

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Figures (13)

Fig. 1
Fig. 1 Wigner distribution function of (a) a coherent state with a p-quadrature mean of 2.5, which corresponds to a number mean of 3.0, and (b) a state whose p- and q-quadratures have been stretched and squeezed, respectively. Both distributions are Gaussian and positive.
Fig. 2
Fig. 2 Noise figure plotted as a function of loss for links (a) without gain and (b) with gain. The dotted blue (green) curve represents a link with a single phase-sensitive (insensitive) amplifier, whereas the solid blue (green) curve represents a link with distributed phase-sensitive (phase-insensitive) amplification. Links with distributed amplification perform significantly better than links with lumped amplification.
Fig. 3
Fig. 3 Fidelity of attenuation plotted as a function of the loss factor. The dotted, dashed, dot-dashed and solid orange curves represent input coherent states with photon-number means of 1, 3, 10 and 30, respectively, whereas the dotted red curve represents an input number state with 1 photon. For most input states, the fidelity decreases rapidly as the loss increases.
Fig. 4
Fig. 4 Fidelity of transmission plotted as a function of link loss. The solid blue and green curves represent links with distributed phase-sensitive and phase-insensitive amplification respectively, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. The inputs to these links are coherent states. The dotted orange and red curves are the attenuation fidelities of 1-photon-mean coherent-state and 1-photon number-state inputs, respectively. Phase-sensitive links with coherent-state inputs perform well.
Fig. 5
Fig. 5 Fidelity of transmission plotted as a function of loss for a link with distributed phase-sensitive amplification. The dotted, dashed, dot-dashed and solid purple curves represent inputs with stretching parameters of 0.1, 0.3, 3, and 10, respectively, whereas the solid blue curve represents an unstretched (coherent-state) input. Stretched inputs perform slightly better than the unstretched input for high losses The solid black curve represents the maximal fidelity (displaced upward by 0.01 for clarity).
Fig. 6
Fig. 6 Purity of the output state plotted as a function of loss for links (a) without gain and (b) with gain. The dotted orange and red curves are the output purities of 1-photon-mean coherent-state and 1-photon number-state inputs, respectively. An attenuated coherent state is pure. The solid blue and green curves represent links with distributed phase-sensitive and phase-insensitive amplification respectively, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. The inputs to these links are coherent states. Phase-sensitive links degrade the purity less than phase-insensitive links, and a stretched coherent state is pure.
Fig. 7
Fig. 7 Purity of the output state plotted as a function of loss for a link with distributed phase-sensitive amplification. The dotted, dashed, dot-dashed and solid purple curves represent inputs with stretching parameters of 0.1, 0.3, 3, and 10, respectively, whereas the solid blue curve represents an unstretched (coherent-state) input. For high losses, the output purity depends only weakly on the stretching parameter.
Fig. 8
Fig. 8 (a) Fidelity of two-mode attenuation plotted as a function of loss. The dotted, dashed, dot-dashed and solid orange curves represent input coherent states with photon-number means of 1, 3, 10 and 30, respectively, whereas the dotted red curve represents an input number state with 1 photon. (b) Fidelity of transmission plotted as a function of loss. The solid blue curve represents a link with distributed phase-sensitive amplification, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. Two-mode phase-sensitive links with coherent-state inputs perform well.
Fig. 9
Fig. 9 Density matrices in the number-state representation. Darker squares denote higher values of |ρmn|, whereas lighter squares denote lower values. (a) input coherent state with a number mean of 3.0, (b) output coherent state produced by a 3-dB attenuator, (c) output state produced by a balanced phase-insensitive link with 3 dB of loss and lumped gain, and (d) output state produced by a balanced phase-sensitive link. The former link degrades the coherence of the state more than the latter.
Fig. 10
Fig. 10 Number distribution of (a) an input coherent state with a number mean of 3.0, (b) the output coherent state produced by a 3-dB attenuator, (c) the output state produced by a balanced phase-insensitive link with 3 dB of loss and lumped gain, and (d) the output state produced by a balanced phase-sensitive link. The former link distorts the number distribution more than the latter.
Fig. 11
Fig. 11 Density matrices in the number-state representation. Darker squares denote higher values of |ρmn|, whereas lighter squares denote lower values. The output states are produced by balanced links with (a) 10 dB of loss and lumped gain, and (b) 10 dB of distributed loss and phase-sensitive gain. The state produced by the former link is nearly incoherent, whereas the state produced by the latter is nearly coherent.
Fig. 12
Fig. 12 Number distributions of the output states produced by balanced links with (a) 10 dB of loss and lumped phase-insensitive gain, and (b) 10 dB of distributed loss and phase-sensitive gain. The former link distorts the number distribution much more than the latter.
Fig. 13
Fig. 13 Wigner distribution functions of number states with 0, 1, 2 and 3 photons. The vacuum distribution (a) is Gaussian and positive, whereas the other distributions are non-Gaussian, and have both positive and negative values. As the photon number increases, so also does the number of oscillations.

Equations (115)

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p m q n = W ( p , q ) p m q n d p d q .
C ( k , l ) = W ( p , q ) exp [ i ( k p + l q ) ] d p d q .
p m q n = lim k 0 lim l 0 ( i k ) m ( i l ) n C ( k , l ) .
W ( p , q ) = C ( k , l ) exp [ i ( k p + l q ) ] d k d l / ( 2 π ) 2 .
C ( k , l ) = exp [ i ( k p ^ + l q ^ ) ] = tr { ρ ^ exp [ i ( k p ^ + l q ^ ) ] } ,
( k p ^ + l q ^ ) n = lim θ 0 ( i θ ) n C ( θ k , θ l ) .
( k p ^ + l q ^ ) n = W ( p , q ) ( k p + l q ) n d p d q .
( p ^ m q ^ n ) s = W ( p , q ) p m q n d p d q ,
W ( X ) = ( Δ s 1 / 2 / 2 π ) exp [ ( X V ) t S ( X V ) / 2 ] ,
exp ( X t A X / 2 ) d 2 x = 2 π / Δ a 1 / 2 ,
X = V , ( X V ) ( X V ) t = S 1 = C .
F 12 = [ tr ( ρ ^ 1 1 / 2 ρ ^ 2 ρ ^ 1 1 / 2 ) 1 / 2 ] 2 = [ tr ( ρ ^ 2 1 / 2 ρ ^ 1 ρ ^ 2 1 / 2 ) 1 / 2 ] 2 .
tr ( ρ ^ 1 ρ ^ 2 ) = 2 π W 1 ( p , q ) W 2 ( p , q ) d p d q ,
F 12 = [ ( Δ 1 Δ 2 ) 1 / 2 / 2 π ] exp [ ( X V 1 ) t S 1 ( X V 1 ) / 2 ] × exp [ ( X V 2 ) t S 2 ( X V 2 ) / 2 ] d 2 x .
( X V ) t S 1 ( X V ) + ( X + V ) t S 2 ( X + V ) = X t ( S 1 + S 2 ) X X t ( S 1 S 2 ) V V t ( S 1 S 2 ) X + V t ( S 1 + S 2 ) V = X t S + X X t S V V t S X + V t S + V = ( X W ) t S + ( X W ) W t S + W + V t S + V = ( X W ) t S + ( X W ) + V t ( S + S S + 1 S ) V ,
F 12 = ( Δ 1 Δ 2 / Δ + ) 1 / 2 exp [ V t ( S + S S + 1 S ) V / 2 ] .
F 12 = D 12 1 / 2 exp [ ( V 1 V 2 ) C 12 1 ( V 1 V 2 ) / 2 ] ,
det ( S 1 + S 2 ) / det ( S 1 S 2 ) = det ( S 1 + S 2 ) det ( S 1 1 S 2 1 ) = det ( S 1 1 + S 2 1 ) ,
S + S S + 1 S = 4 ( S 1 1 + S 2 1 ) 1 ,
ρ ^ = m n | m ρ m n n | ,
ρ m n = 2 π W n m ( p , q ) W ( p , q ) d p d q ,
W n m ( p , q ) = { ( 1 ) n ( 2 m n n ! / m ! ) 1 / 2 ( q + i p ) m n exp ( p 2 q 2 ) L n m n ( 2 p 2 + 2 q 2 ) / π , ( 1 ) m ( 2 n m m ! / n ! ) 1 / 2 ( q i p ) n m exp ( p 2 q 2 ) L m n m ( 2 p 2 + 2 q 2 ) / π ,
p ^ = | τ | p ^ + | ρ | w ^ p , q ^ = | τ | q ^ + | ρ | w ^ q ,
p ^ = ( | μ | + | ν | ) p ^ , q ^ = ( | μ | | ν | ) q ^ ,
p ^ 1 = | μ | p ^ 1 + | ν | p ^ 2 , q ^ 1 = | μ | q ^ 1 | ν | q ^ 2 ,
p ^ 2 = | μ | p ^ 2 + | ν | p ^ 1 , q ^ 2 = | μ | q ^ 2 | ν | q ^ 1 .
p ^ = ( λ + τ ) p ^ + ( λ + ρ ) w ^ p , q ^ = ( λ τ ) q ^ + ( λ ρ ) w ^ q ,
v p = ( λ + τ ) 2 v p + ( λ + ρ ) 2 σ , v q = ( λ τ ) 2 v q + ( λ ρ ) 2 σ ,
v p = v p + σ ( l 1 ) , v q = v q / l 2 + σ ( l 1 ) / l 2 ,
v p = v p + σ s ( l 1 ) , v q = v q / l 2 s + σ ( 1 1 / l 2 s ) / ( l + 1 ) .
d z v p = ( γ α ) v p + α σ , d z v q = ( γ + α ) v q + α σ .
v p ( z ) = v p ( 0 ) exp ( λ p z ) + σ α [ exp ( λ p z ) 1 ] / λ p ,
v q ( z ) = v q ( 0 ) exp ( λ q z ) + σ α [ 1 exp ( λ q z ) ] / λ q ,
v p ( z ) = v p ( 0 ) + σ α z ,
v q ( z ) = v q ( 0 ) exp ( 2 α z ) + σ [ 1 exp ( 2 α z ) ] / 2 .
p ^ 1 = ( μ τ ) p ^ 1 + ( μ ρ ) w ^ p + ν p ^ 2 , q ^ 1 = ( μ τ ) q ^ 1 + ( μ ρ ) w ^ q ν q ^ 2 .
v p = ( μ τ ) 2 v p + ( μ 2 ρ 2 + ν 2 ) σ ,
v p = v p + 2 ( l 1 ) σ .
v p = v p + 2 s ( l 1 ) σ .
d z v p = ( γ α ) v p + ( γ + α ) σ .
v p ( z ) = v p ( 0 ) exp ( λ z ) + σ λ + [ exp ( λ z ) 1 ] / λ ,
v p ( z ) = v p ( 0 ) + 2 σ α z .
p ^ 1 = ( μ τ ) p ^ 1 + ( ν τ ) p ^ 2 + ( μ ρ ) w ^ 1 + ( ν ρ ) w ^ 2 ,
p ^ 2 = ( μ τ ) p ^ 2 + ( ν τ ) p ^ 1 + ( μ ρ ) w ^ 2 + ( ν ρ ) w ^ 1 ,
v 1 = μ 2 τ 2 v 1 + 2 μ ν τ 2 c 12 + ν 2 τ 2 v 2 + ( μ 2 + ν 2 ) ρ 2 σ ,
c 12 = μ ν τ 2 v 1 + ( μ 2 + ν 2 ) τ 2 c 12 + μ ν τ 2 v 2 + 2 μ ν ρ 2 σ ,
v 2 = ν 2 τ 2 v 1 + 2 μ ν τ 2 c 12 + μ 2 τ 2 v 2 + ( μ 2 + ν 2 ) ρ 2 σ ,
v j = v 0 ( 1 + 1 / l 2 ) / 2 + σ ( l + 1 / l ) ( 1 1 / l ) / 2 ,
c 12 = v 0 ( 1 1 / l 2 ) / 2 + σ ( l 1 / l ) ( 1 1 / l ) / 2 ,
v j = v 0 ( 1 + 1 / l 2 s ) / 2 + σ [ s ( l 1 ) + ( 1 1 / l 2 s ) / ( 1 + l ) ] / 2 ,
c 12 = v 0 ( 1 1 / l 2 s ) / 2 + σ [ s ( l 1 ) ( 1 1 / l 2 s ) / ( 1 + l ) ] / 2 .
d z v 1 = α v 1 + γ c 12 + α σ ,
d z c 12 = α c 12 + γ ( v 1 + v 2 ) / 2 ,
d z v 2 = α v 2 + γ c 12 + α σ .
d z s 12 = α s 12 + γ c 12 + α σ ,
d z c 12 = α c 12 + γ s 12 ,
d z d 12 = α d 12 .
s 12 ( z ) = [ s 12 ( 0 ) cosh ( γ z ) + c 12 ( 0 ) sinh ( γ z ) ] exp ( α z ) + σ α { [ α cosh ( γ z ) + γ sinh ( γ z ) ] exp ( α z ) α } / ( γ 2 α 2 ) ,
c 12 ( z ) = [ s 12 ( 0 ) sinh ( γ z ) + c 12 ( 0 ) cosh ( γ z ) ] exp ( α z ) + σ α { [ γ cosh ( γ z ) + α sinh ( γ z ) ] exp ( α z ) γ } / ( γ 2 α 2 ) .
s 12 ( z ) = α [ 2 α z + 3 + exp ( 2 α z ) ] / 4 ,
c 12 ( z ) = σ [ 2 α z + 1 exp ( 2 α z ) ] / 4 ,
F 12 = exp [ ( p 1 2 + q 1 2 ) ( 1 τ ) 2 / 2 ] .
F 12 = τ 2 = 1 / L ,
F 12 = { 8 / [ ( 2 + log L ) ( 3 + 1 / L 2 ) ] } 1 / 2 .
F 12 = 1 / ( 1 + log L ) .
D 12 = 1 + ( λ + 1 / λ ) α z + ( α z ) 2 .
D 12 1 / 2 + λ / 4 + α z / 4 λ + α z / 8 .
C 1 = σ [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] , C 2 = σ [ s 0 c 0 0 s 0 c c 0 s 0 0 c 0 s ] ,
F 12 = 4 / [ ( 1 + s ) 2 c 2 ] .
C ( k , l ) = q | ρ ^ exp [ i ( k p ^ + l q ^ ) ] | q d q .
exp ( x ^ + y ^ ) = exp ( x ^ ) exp ( y ^ ) exp ( [ x ^ , y ^ ] / 2 ) ,
exp [ i ( k p ^ + l q ^ ) ] = exp ( i k p ^ ) exp ( i l q ^ ) exp ( i k l / 2 ) = exp ( i l q ^ ) exp ( i k p ^ ) exp ( i k l / 2 ) .
exp ( i k p ^ ) | p = exp ( i k p ) | p , exp ( i k p ^ ) | q = | q + k ,
exp ( i l q ^ ) | p = | p l , exp ( i l q ^ ) | q = exp ( i l q ) | q .
[ q ^ , p ^ m ] = i m p ^ m 1 , [ p ^ , q ^ n ] = i n q ^ n 1 ,
C ( k , l ) = q k / 2 | ρ ^ | q + k / 2 exp ( i l q ) d q .
W ( p , q ) = q k / 2 | ρ ^ | q + k / 2 exp ( i k p ) d k / 2 π ,
W ( p , q ) = p + l / 2 | ρ ^ | p l / 2 exp ( i l q ) d l / 2 π .
P ( p ) = W ( p , q ) d q = p | ρ ^ | p , Q ( q ) = W ( p , q ) d p = q | ρ ^ | q
W n m ( p , q ) = ψ n ( q k / 2 ) ψ m * ( q + k / 2 ) exp ( i k p ) d k / 2 π ,
ψ n ( q ) = q | n = H n ( q ) exp ( q 2 / 2 ) / ( 2 n n ! ) 1 / 2 π 1 / 4 .
W n m ( p , q ) = [ exp ( p 2 q 2 ) / ( 2 m + n m ! n ! ) 1 / 2 π 3 / 2 ] × H m ( q + i p + l ) H n ( q i p l ) exp ( l 2 ) d l .
H m ( x + y ) H n ( x + z ) exp ( x 2 ) d x = 2 m π 1 / 2 n ! y m n L n m n ( 2 y z )
W n m ( p , q ) = ( 1 ) n ( n ! / m ! ) 1 / 2 [ 2 1 / 2 ( q + i p ) ] m n exp ( p 2 q 2 ) L n m n ( 2 p 2 + 2 q 2 ) / π .
W n m ( p , q ) = ( 1 ) m ( m ! / n ! ) 1 / 2 [ 2 1 / 2 ( q i p ) ] n m exp ( p 2 q 2 ) L m n m ( 2 p 2 + 2 q 2 ) / π ,
q k 1 / 2 | ρ ^ 1 | q + k 1 / 2 q k 2 / 2 | ρ ^ 2 | q + k 2 / 2 exp [ i ( k 1 + k 2 ) p ] d k 1 d k 2 d p d q / 2 π = q k 1 / 2 | ρ ^ 1 | q + k 1 / 2 q k 2 / 2 | ρ ^ 2 | q + k 2 / 2 δ ( k 1 + k 2 ) d k 1 d k 2 d q = q k / 2 | ρ ^ 1 | q + k / 2 q + k / 2 | ρ ^ 2 | q k / 2 d k d q = q | ρ ^ 1 ρ ^ 2 | q d q ,
( A + B ) ( A B ) S 1 ( A B ) = 4 ( A 1 + B 1 ) 1 ,
( A 1 + B 1 ) [ A + B ( A B ) S 1 ( A B ) ] = ( A 1 + B 1 ) [ ( A A S 1 A ) + ( B B S 1 B ) + A S 1 B + B S 1 A ] .
( A 1 + B 1 ) ( A A S 1 A ) = I S 1 A + B 1 A B 1 ( S B ) S 1 A = I .
( A 1 + B 1 ) A S 1 B = S 1 B + B 1 ( S B ) S 1 B = I .
D ^ ( ξ ) = exp ( ξ a ^ ) exp ( ξ * a ^ ) exp ( | ξ | 2 / 2 ) = exp ( ξ * a ^ ) exp ( ξ a ^ ) exp ( | ξ | 2 / 2 ) ,
m | exp ( ξ a ^ ) = k = 0 m ( m k ) 1 / 2 ξ k m k | ( k ! ) 1 / 2 ,
exp ( ξ * a ^ ) | n = l = 0 n ( n l ) 1 / 2 ( ξ * ) l | n l ( l ! ) 1 / 2 .
C n m ( ξ ) = ξ d exp ( | ξ | 2 / 2 ) l = 0 n [ n ! ( n + d ) ! ] 1 / 2 ( | ξ | 2 ) l l ! ( l + d ) ! ( n l ) ! .
L n m ( x ) = l = 0 n ( n + m l + m ) ( x ) l l ! .
C n m ( ξ ) = ( n ! / m ! ) 1 / 2 ξ m n exp ( | ξ | 2 / 2 ) L n m n ( | ξ | 2 ) .
C n m ( ξ ) = ( m ! / n ! ) 1 / 2 ( ξ * ) n m exp ( | ξ | 2 / 2 ) L m n m ( | ξ | 2 ) .
W n m ( α ) = C n m ( ξ ) exp ( α ξ * α * ξ ) d 2 ξ / π ,
C n m ( α ) = W n m ( ξ ) exp ( ξ α * ξ * α ) d 2 α / π ,
W n m ( α ) = ( n ! / m ! ) 1 / 2 ρ d + 1 exp ( ρ 2 / 2 ) L n d ( ρ 2 ) × exp [ i d ϕ + i 2 r ρ sin ( θ ϕ ) ] d ρ d ϕ / π ,
exp ( i d θ ) exp [ i d ( ϕ θ ) i 2 r ρ sin ( ϕ θ ) ] d ( ϕ θ ) .
J p ( x ) = π π exp [ i ( p t x sin t ) ] d t / 2 π ,
W ( r , θ ) = 2 ( n ! / m ! ) 1 / 2 exp ( i d θ ) ρ d + 1 exp ( ρ 2 / 2 ) L n d ( ρ 2 ) J d ( ρ y ) d ρ ,
x d + 1 exp ( x 2 / 2 ) L n d ( x 2 ) J d ( x y ) d x = ( 1 ) n y d exp ( y 2 / 2 ) L n d ( y 2 ) ,
W n m ( r , θ ) = 2 ( 1 ) n ( n ! / m ! ) 1 / 2 ( 2 r ) d exp ( i d θ 2 r 2 ) L n d ( 4 r 2 ) .
W n m ( α ) = 2 ( 1 ) n ( n ! / m ! ) 1 / 2 ( 2 α ) m n exp ( 2 | α | 2 ) L n m n ( 4 | α | 2 ) .
W n m ( α ) = 2 ( 1 ) m ( m ! / n ! ) 1 / 2 ( 2 α * ) n m exp ( 2 | α | 2 ) L m n m ( 4 | α | 2 ) .
W n m ( α ) | m < n = C m n ( ξ * ) | n > m exp ( α ξ * α * ξ ) d 2 ξ / π = C m n ( ξ * ) | n > m exp [ α * ( ξ * ) * α ( ξ * ) ] d 2 ( ξ * ) / π = W m n ( α * ) | n > m .
C r ( k , l ) = C c ( k / 2 1 / 2 , l / 2 1 / 2 ) , W r ( x , y ) = W c ( y / 2 1 / 2 , x / 2 1 / 2 ) / 2 π ,
tr ( ρ 1 ρ 2 ) = W 1 ( α ) W 2 ( α ) d 2 α / π ,
p ^ q ^ = i ( a ^ ) 2 a ^ 2 + 1 / 2 ,
a ^ 1 = τ a ^ + ρ w ^ , a ^ 2 = ρ * a ^ + τ * w ^ ,
p ^ 1 = [ τ ( a ^ + a ^ ) + ρ ( w ^ + w ^ ) ] / 2 1 / 2 , q ^ 2 = i [ ρ ( a ^ a ^ ) + τ ( w ^ + w ^ ) ] / 2 1 / 2 .
p ^ 1 q ^ 2 = i τ ρ ( a ^ ) 2 a ^ 2 / 2 = τ ρ ( p ^ q ^ ) s ,
( p ^ 1 ) 2 = τ 2 p ^ 2 + ρ 2 / 2 , ( q ^ 2 ) 2 = ρ 2 q ^ 2 + τ 2 / 2

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