Communications in Mathematical Physics

, Volume 344, Issue 3, pp 797–829 | Cite as

Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication



This paper studies the difficulty of discriminating between an arbitrary quantum channel and a “replacer" channel that discards its input and replaces it with a fixed state. The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein’s lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel \({\mathcal{N}}\) as the optimal Type II error exponent when discriminating between a large number of independent instances of \({\mathcal{N}}\) and an arbitrary “worst-case” replacer channel chosen from the set of all replacer channels.


Quantum Channel Relative Entropy Direct Part Bipartite State Classical Capacity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ando T.: Convexity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26, 203–241 (1979)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Audenaert K.M.R., Nussbaum M., Szkola A., Verstraete F.: Asymptotic error rates in quantum hypothesis testing. Commun. Math. Phys. 279, 251–283 (2008) arXiv:0708.4282 ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Augustin, U.: Noisy channels. Habilitation thesis, Universitat Erlangen-Nurnberg, West Germany (1978)Google Scholar
  4. 4.
    Beigi S.: Sandwiched Rényi divergence satisfies data processing inequality. J.Math. Phys. 54(12), 122202 (2013) arXiv:1306.5920 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bennett C.H., Devetak I., Harrow A.W., Shor P.W., Winter A.: Quantum reverse Shannon theorem and resource tradeoffs for simulating quantum channels. IEEE Trans. Inf. Theory 60(5), 2926–2959 (2014) arXiv:0912.5537 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83(15), 3081–3084 (1999) arXiv:quant-ph/9904023 ADSCrossRefGoogle Scholar
  7. 7.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48, 2637–2655 (2002) arXiv:quant-ph/0106052 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Blahut R.: Hypothesis testing and information theory. IEEE Trans. Inf. Theory 20(4), 405–417 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bowen G.: Quantum feedback channels. IEEE Trans. Inf. Theory 50, 2429–2433 (2004) arXiv:quant-ph/0209076 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Burnashev M.V., Holevo A.S.: On reliability function of quantum communication channel. Probl. Inf. Transm. 34, 97–107 (1998) arXiv:quant-ph/9703013 MathSciNetMATHGoogle Scholar
  11. 11.
    Carlen E.A., Lieb E.H.: A Minkowski type trace inequality and strong subadditivity of quantum entropy II: convexity and concavity. Lett. Math. Phys. 83(2), 107–126 (2008) arXiv:0710.4167 ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Csiszar I., Korner J.: Feedback does not affect the reliability function of a DMC at rates above capacity. IEEE Trans. Inf. Theory 28(1), 92–93 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Devetak I., King C., Junge M., Ruskai M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266(1), 37–63 (2006) arXiv:quant-ph/0506196 ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Duan R., Feng Y., Ying M.: Perfect distinguishability of quantum operations. Phys. Rev. Lett. 103(21), 210501 (2009) arXiv:0908.0119 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dueck G., Korner J.: Reliability function of a discrete memoryless channel at rates above capacity. IEEE Trans. Inf. Theory 25(1), 82–85 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fan K.: Minimax theorems. Proc. Natl. Acad. Sci. USA 39(1), 42–47 (1953)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Frank R.L., Lieb E.H.: Monotonicity of a relative Rényi entropy. J. Math. Phys. 54(12), 122201 (2013) arXiv:1306.5358 ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gupta M., Wilde M.M.: Multiplicativity of completely bounded p-norms implies a strong converse for entanglement-assisted capacity. Commun. Math. Phys. 334(2), 867–887 (2015) arXiv:1310.7028 ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Harrow A.W., Hassidim A., Leung D., Watrous J.: versus non-adaptive strategies for quantum channel discrimination. Phys. Rev. A 81(3), 032339 (2010) arXiv:0909.0256 ADSCrossRefGoogle Scholar
  20. 20.
    Hayashi M.: Quantum Information Theory: An Introduction. Springer, New York (2006)MATHGoogle Scholar
  21. 21.
    Hayashi M.: Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding. Phys. Rev. A 76(6), 062301 (2007) arXiv:quant-ph/0611013 ADSCrossRefGoogle Scholar
  22. 22.
    Hayashi M.: Discrimination of two channels by adaptive methods and its application to quantum system. IEEE Trans. Inf. Theory 55(8), 3807–3820 (2009) arXiv:0804.0686 MathSciNetCrossRefGoogle Scholar
  23. 23.
    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 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hayashi, M., Tomamichel, M.: Correlation detection and an operational interpretation of the Rényi mutual information. In: 2015 IEEE International Symposium on Information Theorey, Hong Kong, pp. 1447–1451 (2015). arXiv:1408:6894
  25. 25.
    Hiai F., Petz D.: The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143(1), 99–114 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44(1), 269–273 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Holevo A.S.: Reliability function of general classical-quantum channel. IEEE Trans. Inf. Theory 46(6), 2256–2261 (2000) arXiv:quant-ph/9907087 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Holevo A.S.: On entanglement assisted classical capacity. J. Math. Phys. 43(9), 4326–4333 (2002) arXiv:quant-ph/0106075 ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jenčová A.: A relation between completely bounded norms and conjugate channels. Commun. Math. Phys. 266(1), 65–70 (2006) arXiv:quant-ph/0601071 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Kneser H.: Sur un téorème fondamental de la théorie des jeux. C. R. Acad. Sci. Paris 234, 2418–2420 (1952)MathSciNetMATHGoogle Scholar
  31. 31.
    Koenig R., Wehner S.: A strong converse for classical channel coding using entangled inputs. Phys. Rev. Lett. 103(7), 070504 (2009) arXiv:0903.2838 ADSCrossRefGoogle Scholar
  32. 32.
    Li K.: Second order asymptotics for quantum hypothesis testing. Ann. Stat. 42(1), 171–189 (2014) arXiv:1208.1400 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lieb E.H.: Convex trace functions and the Wigner–Yanase–Dyson conjecture. Adv. Math. 11, 267–288 (1973)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Lloyd S.: Enhanced sensitivity of photodetection via quantum illumination. Science 321(5895), 1463–1465 (2008) arXiv:0803.2022 ADSCrossRefGoogle Scholar
  35. 35.
    Mosonyi, M., Hiai, F.: On the quantum Rényi relative entropies and related capacity formulas. IEEE Trans. Inf. Theory 57(4), 2474–2487 (2011)Google Scholar
  36. 36.
    Mosonyi M., Ogawa T.: Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Commun. Math. Phys. 334(3), 1617–1648 (2015) arXiv:1309.3228 ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Müller-Lennert M., Dupuis F., Szehr O., Fehr S., Tomamichel M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54(12), 122203 (2013) arXiv:1306.3142 ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Nagaoka, H.: Strong converse theorems in quantum information theory. In: Proceedings of ERATO Workshop on Quantum Information Science, p.33 (2001). [Also appeared in Asymptotic Theory of Quantum Statistical Inference, ed. M. Hayashi, World Scientific, singapore (2005)]Google Scholar
  39. 39.
    Nagaoka, H.: The converse part of the theorem for quantum Hoeffding bound (2006). arXiv:quant-ph/0611289
  40. 40.
    Ogawa T., Hayashi M.: On error exponents in quantum hypothesis testing. IEEE Trans. Inf. Theory 50(6), 1368–1372 (2004) arXiv:quant-ph/0206151 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Ogawa T., Nagaoka H.: Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inf. Theory 46(7), 2428–2433 (2000) arXiv:quant-ph/9906090 MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Ogawa T., Nagaoka H.: Making good codes for classical-quantum channel coding via quantum hypothesis testing. IEEE Trans. Inf. Theory 53(6), 2261–2266 (2007) arXiv:quant-ph/0208139 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Petz D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Pisier, G.: Non-commutative vector valued L p-spaces and completely p-summing maps. Astérisque, vol. 247, Socité Mathématique de France (1998)Google Scholar
  45. 45.
    Polyanskiy, Y., Verdú, S.: Arimoto channel coding converse and Rényi divergence. In: Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computation, pp. 1327–1333 (2010)Google Scholar
  46. 46.
    Renner, R.: Security of quantum key distribution. Ph.D. thesis, ETH Zurich (2005). arXiv:quant-ph/0512258
  47. 47.
    Sacchi M.F.: Entanglement can enhance the distinguishability of entanglement-breaking channels. Phys. Rev. A 72(1), 014305 (2005) arXiv:quant-ph/0505174 ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Sacchi M.F.: Optimal discrimination of quantum operations. Phys. Rev. A 71(6), 062340 (2005) arXiv:quant-ph/0505183 ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Schumacher B., Westmoreland M.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)ADSCrossRefGoogle Scholar
  50. 50.
    Tan S.-H., Erkmen B.I., Giovannetti V., Guha S., Lloyd S., Maccone L., Pirandola S., Shapiro J.H.: Quantum illumination with Gaussian states. Phys. Rev. Lett. 101(25), 253601 (2008) arXiv:0810.0534 ADSCrossRefGoogle Scholar
  51. 51.
    Tomamichel M., Colbeck R., Renner R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55(12), 5840–5847 (2009) arXiv:0811.1221 MathSciNetCrossRefGoogle Scholar
  52. 52.
    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
  53. 53.
    Tomamichel, M., Wilde, M.M., Winter, A.: Strong converse rates for quantum communication. In: 2015 IEEE International Symposium on Information Theorey, Hong Kong, pp. 2386–2390 (2015). arXiv:1406.2946
  54. 54.
    Umegaki H.: Conditional expectation in an operator algebra. Kodai Math. Semin. Rep. 14(2), 59–85 (1962)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Wang L., Renner R.: One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108(20), 200501 (2012) arXiv:1007.5456 ADSCrossRefGoogle Scholar
  56. 56.
    Wilde M.M., Winter A., Yang D.: Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331(2), 593–622 (2014) arXiv:1306.1586 ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Physics and Astronomy, Hearne Institute for Theoretical PhysicsLouisiana State UniversityBaton RougeUSA
  2. 2.Física Teòrica: Informació i Fenòmens QuànticsUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Mathematical InstituteBudapest University of Technology and EconomicsBudapestHungary
  4. 4.Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA

Personalised recommendations