Advertisement

Communications in Mathematical Physics

, Volume 355, Issue 3, pp 1283–1315 | Cite as

Moderate Deviation Analysis for Classical Communication over Quantum Channels

  • Christopher T. ChubbEmail author
  • Vincent Y. F. Tan
  • Marco Tomamichel
Article

Abstract

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altŭg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998) arXiv:quant-ph/9611023 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Holevo A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inf. Transm. 9(3), 177–183 (1973)Google Scholar
  3. 3.
    Schumacher B., Westmoreland M.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)ADSCrossRefGoogle Scholar
  4. 4.
    Holevo A.: Reliability function of general classical-quantum channel. IEEE Trans. Inf. Theory 46, 2256–2261 (2000) arXiv:quant-ph/9907087 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hayashi M.: Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding. Phys. Rev. A 76, 062301 (2006) arXiv:quant-ph/0611013 ADSCrossRefGoogle Scholar
  6. 6.
    Dalai M.: Lower bounds on the probability of error for classical and classical-quantum channels. IEEE Trans. Inf. Theory 59, 8027–8056 (2013) arXiv:1201.5411 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Tomamichel M., Tan V.Y.F.: Second-order asymptotics for the classical capacity of image-additive quantum channels. Commun. Math. Phys. 338, 103–137 (2015) arXiv:1308.6503 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Stochastic Modelling and Applied Probability. Springer, Berlin (1998)Google Scholar
  9. 9.
    Umegaki H.: Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math. Semin. Rep. 14(2), 59–85 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tomamichel M., Hayashi M.: A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans. Inf. Theory 59, 7693–7710 (2013) arXiv:1208.1478 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li K.: Second-order asymptotics for quantum hypothesis testing. Ann. Stat. 42, 171–189 (2014) arXiv:1208.1400 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Altug Y., Wagner A.B.: Moderate deviations in channel coding. IEEE Trans. Inf. Theory 60, 4417–4426 (2014) arXiv:1208.1924 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Polyanskiy, Y., Verdu, S.: Channel dispersion and moderate deviations limits for memoryless channels. In: 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1334–1339, IEEE (2010)Google Scholar
  14. 14.
    Wang L., Renner R.: One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108, 200501 (2012) arXiv:1007.5456 ADSCrossRefGoogle Scholar
  15. 15.
    Cheng, H.-C., Hsieh, M.-H.: Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing (2016). arXiv:1701.03195
  16. 16.
    Hoeffding W.: Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36(2), 369–401 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gallager R.G.: Information Theory and Reliable Communication. Wiley, London (1968)zbMATHGoogle Scholar
  18. 18.
    Csiszár I., Körner J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Nagaoka, H.: The Converse Part of The Theorem for Quantum Hoeffding Bound (2006). arXiv:quant-ph/0611289
  20. 20.
    Sason, I.: Moderate deviations analysis of binary hypothesis testing. In: 2012 IEEE International Symposium on Information Theory Proceedings, pp. 821–825. IEEE (2012). arXiv:1111.1995
  21. 21.
    Strassen, V.: Asymptotische Abschätzungen in Shannons Informationstheorie. In: Trans. Third Prague Conference on Information Theory, Prague, pp. 689–723 (1962)Google Scholar
  22. 22.
    Hayashi M.: Information spectrum approach to second-order coding rate in channel coding. IEEE Trans. Inf. Theory 55, 4947–4966 (2009) arXiv:0801.2242 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Polyanskiy Y., Poor H.V., Verdú S.: Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory 56, 2307–2359 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Csiszár I., Longo G.: On the error exponent for source coding and for testing simple statistical hypotheses. Stud. Sci. Math. Hung. 6, 181–191 (1971)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Han, T.S., Kobayashi, K.: The strong converse theorem for hypothesis testing. IEEE Trans. Inf. Theory 35, 178–180 (1989)Google Scholar
  26. 26.
    Arimoto S.: On the converse to the coding theorem for discrete memoryless channels. IEEE Trans. Inf. Theory 19, 357–359 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Dueck G., Korner J.: Reliability function of a discrete memoryless channel at rates above capacity. (Corresp.). IEEE Trans. Inf. Theory 25, 82–85 (1979)CrossRefzbMATHGoogle Scholar
  28. 28.
    Mosonyi M., Ogawa T.: Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Commun. Math. Phys. 334, 1617–1648 (2015) arXiv:1309.3228 ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Mosonyi M., Ogawa T.: Two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states. IEEE Trans. Inf. Theory 61, 6975–6994 (2014) arXiv:1407.3567 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mosonyi, M., Ogawa, T.: Strong Converse Exponent for Classical-Quantum Channel Coding (2014). arXiv:1409.3562
  31. 31.
    Parthasarathy K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)CrossRefzbMATHGoogle Scholar
  32. 32.
    Winter A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Theory 45, 2481–2485 (2014) arXiv:1409.2536 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ogawa T., Nagaoka H.: Strong converse to the quantum channel coding theorem. IEEE Trans. Inf. Theory 45, 2486–2489 (1999) arXiv:quant-ph/9808063 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Schumacher B., Westmoreland M.D.: Optimal signal ensembles. Phys. Rev. A 63, 022308 (1999) arXiv:quant-ph/9912122 ADSCrossRefGoogle Scholar
  35. 35.
    Dupuis, F., Kraemer, L., Faist, P., Renes, J.M., Renner, R.: Generalized entropies. In: Proceedings of XVIIth International Congress on Mathematical Physics, pp. 134–153 (2012). arXiv:1211.3141
  36. 36.
    Petz D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lin M.S., Tomamichel M.: Investigating properties of a family of quantum Renyi divergences. Quantum Inf. Process. 14, 1501–1512 (2014) arXiv:1408.6897 ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Nussbaum M., Szkoła A.: The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Stat. 37, 1040–1057 (2009) arXiv:quant-ph/0607216 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Fukuda M., Nechita I., Wolf M.M.: Quantum channels with polytopic images and image additivity. IEEE Trans. Inf. Theory 61, 1851–1859 (2015) arXiv:1408.2340 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hayashi M., Nagaoka H.: General formulas for capacity of classical-quantum channels. IEEE Trans. Inf. Theory 49, 1753–1768 (2003) arXiv:quant-ph/0206186 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Hayden P., Leung D., Shor P.W., Winter A.: Randomizing quantum states: constructions and applications. Commun. Math. Phys. 250, 371–391 (2004) arXiv:quant-ph/03071 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Polyanskiy, Y.: Channel coding: non-asymptotic fundamental limits. Ph.D. thesis, Princeton University (2010)Google Scholar
  43. 43.
    Tomamichel M., Tan V.Y.F.: A tight upper bound for the third-order asymptotics for most discrete memoryless channels. IEEE Trans. Inf. Theory 59, 7041–7051 (2013) arXiv:1212.3689 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Moulin P.: The log-volume of optimal codes for memoryless channels, asymptotically within a few nats. IEEE Trans. Inf. Theory 63, 2278–2313 (2017) arXiv:1311.0181 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Datta N., Leditzky F.: Second-order asymptotics for source coding, dense coding, and pure-state entanglement conversions. IEEE Trans. Inf. Theory 61, 582–608 (2015) arXiv:1403.2543 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Leditzky F., Datta N.: Second-order asymptotics of visible mixed quantum source coding via universal codes. IEEE Trans. Inf. Theory 62, 4347–4355 (2016) arXiv:1407.6616 MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Datta N., Tomamichel M., Wilde M.M.: On the second-order asymptotics for entanglement-assisted communication. Quantum Inf. Process. 15, 2569–2591 (2016) arXiv:1405.1797 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tomamichel M., Berta M., Renes J.M.: Quantum coding with finite resources. Nat. Commun. 7, 11419 (2016) arXiv:1504.04617 ADSCrossRefGoogle Scholar
  49. 49.
    Wilde M.M., Tomamichel M., Berta M.: Converse bounds for private communication over quantum channels. IEEE Trans. Inf. Theory 63, 1792–1817 (2017) arXiv:1602.08898 MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Rozovsky L.V.: Estimate from below for large-deviation probabilities of a sum of independent random variables with finite variances. J. Math. Sci. 109(6), 2192–2209 (2002)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lee S.-H., Tan V.Y.F., Khisti A.: Streaming data transmission in the moderate deviations and central limit regimes. IEEE Trans. Inf. Theory 62, 6816–6830 (2016) arXiv:1512.06298 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Christopher T. Chubb
    • 1
    Email author
  • Vincent Y. F. Tan
    • 2
    • 3
  • Marco Tomamichel
    • 1
    • 4
  1. 1.Centre for Engineered Quantum Systems, School of PhysicsUniversity of SydneySydneyAustralia
  2. 2.Department of Electrical and Computer EngineeringNational University of SingaporeSingaporeSingapore
  3. 3.Department of MathematicsNational University of SingaporeSingaporeSingapore
  4. 4.Centre for Quantum Software and InformationUniversity of Technology SydneySydneyAustralia

Personalised recommendations