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Moderate Deviation Analysis for Classical Communication over Quantum Channels

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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.

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References

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. Schumacher B., Westmoreland M.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)

    Article  ADS  Google Scholar 

  4. Holevo A.: Reliability function of general classical-quantum channel. IEEE Trans. Inf. Theory 46, 2256–2261 (2000) arXiv:quant-ph/9907087

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Stochastic Modelling and Applied Probability. Springer, Berlin (1998)

  9. Umegaki H.: Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math. Semin. Rep. 14(2), 59–85 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  11. Li K.: Second-order asymptotics for quantum hypothesis testing. Ann. Stat. 42, 171–189 (2014) arXiv:1208.1400

    Article  MathSciNet  MATH  Google Scholar 

  12. Altug Y., Wagner A.B.: Moderate deviations in channel coding. IEEE Trans. Inf. Theory 60, 4417–4426 (2014) arXiv:1208.1924

    Article  MathSciNet  MATH  Google Scholar 

  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)

  14. Wang L., Renner R.: One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108, 200501 (2012) arXiv:1007.5456

    Article  ADS  Google Scholar 

  15. Cheng, H.-C., Hsieh, M.-H.: Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing (2016). arXiv:1701.03195

  16. Hoeffding W.: Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36(2), 369–401 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gallager R.G.: Information Theory and Reliable Communication. Wiley, London (1968)

    MATH  Google Scholar 

  18. Csiszár I., Körner J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  19. Nagaoka, H.: The Converse Part of The Theorem for Quantum Hoeffding Bound (2006). arXiv:quant-ph/0611289

  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. Strassen, V.: Asymptotische Abschätzungen in Shannons Informationstheorie. In: Trans. Third Prague Conference on Information Theory, Prague, pp. 689–723 (1962)

  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

    Article  MathSciNet  MATH  Google Scholar 

  23. Polyanskiy Y., Poor H.V., Verdú S.: Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory 56, 2307–2359 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  25. Han, T.S., Kobayashi, K.: The strong converse theorem for hypothesis testing. IEEE Trans. Inf. Theory 35, 178–180 (1989)

  26. Arimoto S.: On the converse to the coding theorem for discrete memoryless channels. IEEE Trans. Inf. Theory 19, 357–359 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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

    Article  ADS  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  30. Mosonyi, M., Ogawa, T.: Strong Converse Exponent for Classical-Quantum Channel Coding (2014). arXiv:1409.3562

  31. Parthasarathy K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)

    Book  MATH  Google Scholar 

  32. Winter A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Theory 45, 2481–2485 (2014) arXiv:1409.2536

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  34. Schumacher B., Westmoreland M.D.: Optimal signal ensembles. Phys. Rev. A 63, 022308 (1999) arXiv:quant-ph/9912122

    Article  ADS  Google Scholar 

  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. Petz D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MATH  Google Scholar 

  38. Nussbaum M., Szkoła A.: The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Stat. 37, 1040–1057 (2009) arXiv:quant-ph/0607216

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Polyanskiy, Y.: Channel coding: non-asymptotic fundamental limits. Ph.D. thesis, Princeton University (2010)

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Tomamichel M., Berta M., Renes J.M.: Quantum coding with finite resources. Nat. Commun. 7, 11419 (2016) arXiv:1504.04617

    Article  ADS  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia

    Christopher T. Chubb & Marco Tomamichel

  2. Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore

    Vincent Y. F. Tan

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Vincent Y. F. Tan

  4. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia

    Marco Tomamichel

Authors
  1. Christopher T. Chubb
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  2. Vincent Y. F. Tan
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  3. Marco Tomamichel
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Correspondence to Christopher T. Chubb.

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Communicated by M. M. Wolf

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Chubb, C.T., Tan, V.Y.F. & Tomamichel, M. Moderate Deviation Analysis for Classical Communication over Quantum Channels. Commun. Math. Phys. 355, 1283–1315 (2017). https://doi.org/10.1007/s00220-017-2971-1

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  • DOI: https://doi.org/10.1007/s00220-017-2971-1

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