Quantum Information Processing

, Volume 15, Issue 3, pp 1289–1308 | Cite as

Second-order coding rates for pure-loss bosonic channels

Article

Abstract

A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon-number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest one, the pure-loss bosonic channel. In a second-order analysis of communication, one fixes the tolerable error probability and seeks to understand the back-off from capacity for a sufficiently large yet finite number of channel uses. We find a lower bound on the maximum achievable code size for the pure-loss bosonic channel, in terms of the known expression for its capacity and a quantity called channel dispersion. We accomplish this by proving a general “one-shot” coding theorem for channels with classical inputs and pure-state quantum outputs which reside in a separable Hilbert space. The theorem leads to an optimal second-order characterization when the channel output is finite-dimensional, and it remains an open question to determine whether the characterization is optimal for the pure-loss bosonic channel.

Keywords

Second-order coding rates Pure-loss bosonic channel   Optical communication Spectral entropy 

References

  1. 1.
    Bardhan, B.R., Garcia-Patron, R., Wilde, M.M., Winter, A.: Strong converse for the classical capacity of optical quantum communication channels. (2014). arXiv:1401.4161
  2. 2.
    Bardhan, B.R., Wilde, M.M.: Strong converse rates for classical communication over thermal and additive noise bosonic channels. Phys. Rev. A 89(2), 022302 (2014). arXiv:1312.3287
  3. 3.
    Belavkin, V.: Optimal distinction of non-orthogonal quantum signals. Radio Eng. Electron. Phys. 20, 39–47 (1975)ADSGoogle Scholar
  4. 4.
    Belavkin, V.: Optimal multiple quantum statistical hypothesis testing. Stochastics 1, 315–345 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burnashev, M.V., Holevo, A.S.: On reliability function of quantum communication channel. Probl. Inf. Transm. 34, 97–107 (1998). arXiv:quant-ph/9703013
  6. 6.
    Datta, N., Leditzky, F.: Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions. (2014). arXiv:1403.2543
  7. 7.
    Datta, N., Tomamichel, M., Wilde, M.M.: Second-order coding rates for entanglement-assisted communication. (2014). arXiv:1405.1797
  8. 8.
    Dolinar, S.: A class of optical receivers using optical feedback. PhD thesis, Massachusetts Institute of Technology, June (1976)Google Scholar
  9. 9.
    Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971)MATHGoogle Scholar
  10. 10.
    Gerry, C., Knight, P.: Introductory Quantum Optics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  11. 11.
    Giovannetti, V., Guha, S., Lloyd, S., Maccone, L., Shapiro, J.H., Yuen, H.P.: Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92(2), 027902 (2004). arXiv:quant-ph/0308012
  12. 12.
    Giovannetti, V., Holevo, A.S., García-Patrón, R.: A solution of the Gaussian optimizer conjecture. (2013). arXiv:1312.2251
  13. 13.
    Giovannetti, V., Lloyd, S., Maccone, L.: Achieving the Holevo bound via sequential measurements. Phys. Rev. A 85(1), 012302 (2012). arXiv:1012.0386 ADSCrossRefGoogle Scholar
  14. 14.
    Gordon, J.P.: Noise at optical frequencies; information theory. In: Miles, P.A. (ed.) Quantum Electronics and Coherent Light; Proceedings of the International School of Physics Enrico Fermi, Course XXXI, pp. 156–181. Academic Press, New York (1964)Google Scholar
  15. 15.
    Hausladen, P., Jozsa, R., Schumacher, B., Westmoreland, M., Wootters, W.K.: Classical information capacity of a quantum channel. Phys. Rev. A 54(3), 1869–1876 (1996)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hayashi, M.: Second-order asymptotics in fixed-length source coding and intrinsic randomness. IEEE Trans. Inf. Theory 54(10), 4619–4637 (2008). arXiv:cs/0503089
  17. 17.
    Hayashi, M.: Information spectrum approach to second-order coding rate in channel coding. IEEE Trans. Inf. Theory 55(11), 4947–4966 (2009). arXiv:0801.2242 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hayashi, M., Nagaoka, H.: General formulas for capacity of classical-quantum channels. IEEE Trans. Inf. Theory 49(7), 1753–1768 (2003). arXiv:quant-ph/0206186
  19. 19.
    Holevo, Alexander S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inf. Transm. 9(3), 177–183 (1973)Google Scholar
  20. 20.
    Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44(1), 269–273 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Holevo, A.S., Werner, R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63(3), 032312 (2001). arXiv:quant-ph/9912067
  22. 22.
    Kumagai, W., Hayashi, M.: Entanglement concentration is irreversible. Phys. Rev. Lett. 111(13), 130407 (2013). arXiv:1305.6250 ADSCrossRefGoogle Scholar
  23. 23.
    Li, K.: Second order asymptotics for quantum hypothesis testing. Ann. Stat. 42(1), 171–189 (2014). arXiv:1208.1400 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lloyd, S., Giovannetti, V., Maccone, L.: Sequential projective measurements for channel decoding. Phys. Rev. Lett. 106(25), 250501 (2011). arXiv:1012.0106 ADSCrossRefGoogle Scholar
  25. 25.
    Mari, A., Giovannetti, V., Holevo, A.S.: Quantum state majorization at the output of bosonic Gaussian channels. Nat. Commun. 5, 3826 (2014). arXiv:1312.3545 ADSCrossRefGoogle Scholar
  26. 26.
    Polyanskiy, Y., Poor, H.V., Verdú, S.: Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory 56(5), 2307–2359 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)ADSCrossRefGoogle Scholar
  28. 28.
    Sen, P.: Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding. (2011). arXiv:1109.0802
  29. 29.
    Sen, P.: June 2014. private communicationGoogle Scholar
  30. 30.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Shapiro, J.H.: The quantum theory of optical communications. IEEE J. Sel. Top. Quantum Electron. 15(6), 1547–1569 (2009)CrossRefGoogle Scholar
  32. 32.
    Strassen, V.: Asymptotische Abschätzungen in Shannons Informationstheorie. In: Trans. Third Prague Conf. Inf. Theory, pp. 689–723, Prague (1962)Google Scholar
  33. 33.
    Tan, V.Y.F.: Asymptotic estimates in information theory with non-vanishing error probabilities. Found. Trends Commun. Inf. Theory 11(1–2), 1–184 (2014)ADSCrossRefMATHGoogle Scholar
  34. 34.
    Tan, V.Y.F., Tomamichel, M.: The third-order term in the normal approximation for the AWGN channel. Accepted for publication in IEEE Transactions on Information Theory (2015). arXiv:1311.2337
  35. 35.
    Tomamichel, M., Hayashi, M.: A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans. Inf. Theory 59(11), 7693–7710 (2013). arXiv:1208.1478 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tomamichel, M., Tan, V.Y.F.: Second-order asymptotics for the classical capacity of image-additive quantum channels. Accepted for publication in Communications in Mathematical Physics, August (2013). arXiv:1308.6503
  37. 37.
    Tyurin, I.S.: An improvement of upper estimates of the constants in the Lyapunov theorem. Russ. Math. Surv. 65(3), 201–202 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wang, L., Renner, R.: One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108(20), 200501 (2012). arXiv:1007.5456 ADSCrossRefGoogle Scholar
  39. 39.
    Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information. Rev. Mod. Phys. 84(2), 621–669 (2012). arXiv:1110.3234 ADSCrossRefGoogle Scholar
  40. 40.
    Wilde, M.M.: From classical to quantum Shannon theory. (2011). arXiv:1106.1445
  41. 41.
    Wilde, M.M.: Quantum Inf. Theory. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  42. 42.
    Wilde, M.M., Guha, S., Tan, S.-H., Lloyd, S.: Explicit capacity-achieving receivers for optical communication and quantum reading. In: Proceedings of the 2012 International Symposium on Information Theory, pp. 551–555, Boston, Massachusetts, USA (2012). arXiv:1202.0518
  43. 43.
    Wilde, M.M., Winter, A.: Strong converse for the classical capacity of the pure-loss bosonic channel. Probl. Inf. Transm. 50(2), 117–132 (2014). arXiv:1308.6732 MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Yuen, H.P., Ozawa, M.: Ultimate information carrying limit of quantum systems. Phys. Rev. Lett. 70(4), 363–366 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics and Astronomy, Hearne Institute for Theoretical Physics, Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  3. 3.Quantum Information Processing GroupRaytheon BBN TechnologiesCambridgeUSA

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