Communications in Mathematical Physics

, Volume 331, Issue 2, pp 593–622 | Cite as

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

  • Mark M. WildeEmail author
  • Andreas Winter
  • Dong Yang


A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a “sandwiched” Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.


Success Probability Quantum Channel Density Operator Relative Entropy Quantum Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amosov, G.G., Holevo, A.S., Werner, R.F.: On some additivity problems in quantum information theory. Probl. Inform. Transm. 36(4), 25 (2000). arXiv:math-ph/0003002
  2. 2.
    Arimoto S.: On the converse to the coding theorem for discrete memoryless channels. IEEE Trans. Inform. Theory 19, 357–359 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beigi, S.: Sandwiched Rényi divergence satisfies data processing inequality. J. Math. Phys. 54(12), 122202 (2013). arXiv:1306.5920
  4. 4.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A.:Capacities of quantum erasure channels. Phys. Rev. Lett. 78(16), 3217–3220 (1997) arXiv:quant-ph/9701015
  5. 5.
    Berta, M., Renes, J.M., Wilde, M.M.: Identifying the information gain of a quantum measurement. (2013). arXiv:1301.1594
  6. 6.
    Brádler, K.: An infinite sequence of additive channels: the classical capacity of cloning channels. IEEE Trans. Inform. Theory 57(8), 5497–5503 (2011). arXiv:0903.1638
  7. 7.
    Brádler, K., Dutil, N., Hayden, P., Muhammad, A.: Conjugate degradability and the quantum capacity of cloning channels. J. Math. Phys. 51(7), 072201 (2010). arXiv:0909.3297
  8. 8.
    Brádler, K., Hayden, P., Panangaden, P.: Private information via the Unruh effect. J. High Energy Phys. 2009(08), 074 (2009). arXiv:0807.4536
  9. 9.
    Brito, F., DiVincenzo, D.P., Koch, R.H., Steffen, M.: Efficient one- and two-qubit pulsed gates for an oscillator-stabilized Josephson qubit. New J. Phys. 10(3), 033027 (2008)Google Scholar
  10. 10.
    Buscemi, F., Hayashi, M., Horodecki, M.: Global information balance in quantum measurements. Phys. Rev. Lett. 100, 210504 (2008). arXiv:quant-ph/0702166
  11. 11.
    Busch P.: Informationally complete sets of physical quantities. Int. J. Theor. Phys. 30(9), 1217–1227 (1991)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Carlen E.A.: Trace inequalities and quantum entropy: an introductory course. Contemp. Math. 529, 73–140 (2010)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Carlen, E.A., Lieb, E.H.: A Minkowski type trace inequality and strong subadditivity of the quantum entropy II. Lett. Math. Phys. 83(2), 107–126 (2008) arXiv:0710.4167
  14. 14.
    Chiribella, G.: On quantum estimation, quantum cloning and finite quantum de finetti theorems. In: Theory of Quantum Computation, Communication, and Cryptography. Lecture Notes in Computer Science, vol. 6519, pp. 9–25 (2011). arXiv:1010.1875
  15. 15.
    Csiszár I.: Generalized cutoff rates and Rényi’s information measures. IEEE Trans. Inform. Theory 41(1), 26–34 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dall’Arno, M., D’Ariano, G.M., Sacchi, M.F.: Informational power of quantum measurements. Phys. Rev. A 83, 062304 (2011). arXiv:1103.1972
  17. 17.
    Datta, N., Holevo, A.S., Suhov, Y.: Additivity for transpose depolarizing channels. Int. J. Quantum Infor. 4(1), 85–98 (2006). arXiv:quant-ph/0412034
  18. 18.
    Devetak, I., Shor, P.W.: The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256, 287–303 (2005). arXiv:quant-ph/0311131
  19. 19.
    Dupuis, F., Fawzi, O., Wehner, S.: Entanglement sampling and applications. May 2013. arXiv:1305.1316
  20. 20.
    Dupuis, F., Szehr, O., Tomamichel, M.: unpublished notes (2013)Google Scholar
  21. 21.
    El Gamal A., Kim Y.-H.: Network Information Theory. Cambridge University Press, Cambridge (2012)Google Scholar
  22. 22.
    Fehr, S.: On the conditional Rényi entropy. In: Lecture at the Beyond IID Workshop at the University of Cambridge, January (2013)Google Scholar
  23. 23.
    Frank, R.L., Lieb, E.H.: Monotonicity of a relative Rényi entropy. J. Math. Phys. 54(12), 122201 (2013). arXiv:1306.5358
  24. 24.
    Fukuda, M.: Extending additivity from symmetric to asymmetric channels. J. Phys. A Math. General 38(45), L753–L758 (2005). arXiv:quant-ph/0505022
  25. 25.
    Gallager R.G.: Information Theory and Reliable Communication. Wiley, New York (1968)zbMATHGoogle Scholar
  26. 26.
    Gupta, M.K., Wilde, M.M.: Multiplicativity of completely bounded p-norms implies a strong converse for entanglement-assisted capacity. October (2013). arXiv:1310.7028
  27. 27.
    Gurvits, L., Barnum, H.: Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A 66(6), 062311 (2002). arXiv:quant-ph/0204159
  28. 28.
    Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255–257 (2009). arXiv:0809.3972
  29. 29.
    Holevo A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inform. Transm. 9, 177–183 (1973)MathSciNetGoogle Scholar
  30. 30.
    Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans Inform Theory 44(1), 269–273 (1998). arXiv:quant-ph/9611023
  31. 31.
    Holevo A.S.: Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1999)CrossRefGoogle Scholar
  32. 32.
    Holevo A.S.: Multiplicativity of p-norms of completely positive maps and the additivity problem in quantum information theory. Russ. Math. Surv. 61(2), 301–339 (2006)CrossRefzbMATHGoogle Scholar
  33. 33.
    Holevo, A.S.: Information capacity of quantum observable. Probl. Inform. Transm. 48, 1 (2012) arXiv:1103.2615
  34. 34.
    Horodecki, M., Shor, P.W., Ruskai, M.B.: Entanglement breaking channels. Rev. Math. Phys. 15(6), 629–641 (2003) arXiv:quant-ph/0302031
  35. 35.
    Jacobs, K.: On the properties of information gathering in quantum and classical measurements (2003). arXiv:quant-ph/0304200v1
  36. 36.
    King, C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4653 (2002). arXiv:quant-ph/0103156
  37. 37.
    King, C.: An application of the Lieb-Thirring inequality in quantum information theory. In: Fourteenth International Congress on Mathematical Physics, pp. 486–490 (2003). arXiv:quant-ph/0412046
  38. 38.
    King, C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inform. Theory 49(1), 221–229 (2003). arXiv:quant-ph/0204172
  39. 39.
    King, C.: Maximal p-norms of entanglement breaking channels. Quantum Inform. Comput. 3(2), 186–190 (2003). arXiv:quant-ph/0212057
  40. 40.
    King, C., Matsumoto, K., Nathanson, M., Ruskai, M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. Markov Process. Relat. Fields 13(2), 391–423 (2007). J. T. Lewis memorial issue, arXiv:quant-ph/0509126
  41. 41.
    Koenig, R., Wehner, S.: A strong converse for classical channel coding using entangled inputs. Phys. Rev. Lett. 103, 070504 (2009). arXiv:0903.2838
  42. 42.
    Koenig, R., Wehner, S., Wullschleger, J.: Unconditional security from noisy quantum storage. IEEE Trans. Inform. Theory, 58(3), 1962–1984 (2012). arXiv:0906.1030
  43. 43.
    Lamas-Linares A., Simon C., Howell J.C., Bouwmeester D.: Experimental quantum cloning of single photons. Science 296, 712–714 (2002)ADSCrossRefGoogle Scholar
  44. 44.
    Leung, D., Smith, G.: Continuity of quantum channel capacities. Commun. Math. Phys. 292(1), 201–215 (2009). arXiv:0810.4931
  45. 45.
    Lieb, E.H., Thirring, W.: Studies in mathematical physics. In: Inequalities for the Moments of the Eigenvalues of the Schroedinger Hamiltonian and their Relation to Sobolev Inequalities, pp. 269–297. Princeton University Press, Princeton (1976)Google Scholar
  46. 46.
    Matthews, W.: A linear program for the finite block length converse of Polyanskiy-Poor-Verdú via nonsignaling codes. IEEE Trans. Inform. Theory, 58(12), 7036–7044 (2012). arXiv:1109.5417
  47. 47.
    Matthews, W., Wehner, S.: Finite blocklength converse bounds for quantum channels. October (2012). arXiv:1210.4722
  48. 48.
    Milonni P.W., Hardies M.L.: Photons cannot always be replicated. Phys. Lett. A 92(7), 321–322 (1982)ADSCrossRefGoogle Scholar
  49. 49.
    Mosonyi, M., Hiai, F.: On the quantum Rényi relative entropies and related capacity formulas. IEEE Trans. Inform Theory 57(4), 2474–2487 (2011). arXiv:0912.1286
  50. 50.
    Müller-Lennert, M.: Quantum relative Rényi entropies. Master’s thesis, ETH Zurich, April (2013)Google Scholar
  51. 51.
    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
  52. 52.
    Ogawa, T., Nagaoka, H.: Strong converse to the quantum channel coding theorem. IEEE Trans. Inform. Theory 45(7), 2486–2489 (1999). arXiv:quant-ph/9808063
  53. 53.
    Ohya M., Petz D., Watanabe N.: On capacities of quantum channels. Probab. Math. Stat. Wroclaw Univ. 17, 179–196 (1997)zbMATHMathSciNetGoogle Scholar
  54. 54.
    Oreshkov, O., Calsamiglia, J., Muñoz-Tapia, R., Bagan, E.: Optimal signal states for quantum detectors. New J. Phys. 13(7), 073032 (2011) arXiv:1103.2365
  55. 55.
    Petz D.: Quasi-entropies for finite quantum systems. Reports Math. Phys. 23, 57–65 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    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
  57. 57.
    Prugovečki E.: Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 16, 321–331 (1977)CrossRefzbMATHGoogle Scholar
  58. 58.
    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 21712180 (2004). arXiv:quant-ph/0310075
  59. 59.
    Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)ADSCrossRefGoogle Scholar
  60. 60.
    Schumacher B., Westmoreland M.D.: Optimal signal ensembles. Phys. Rev. A 63, 022308 (2001)ADSCrossRefGoogle Scholar
  61. 61.
    Sharma, N., Warsi, N.A.: On the strong converses for the quantum channel capacity theorems (2012). arXiv:1205.1712
  62. 62.
    Shor, P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43(9), 4334–4340 (2002). arXiv:quant-ph/0201149
  63. 63.
    Sibson R.: Information radius. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14(2), 149160 (1969)CrossRefMathSciNetGoogle Scholar
  64. 64.
    Simon C., Weihs G., Zeilinger A.: Optimal quantum cloning via stimulated emission. Phys. Rev. Lett. 84(13), 2993–2996 (2000)ADSCrossRefGoogle Scholar
  65. 65.
    Sion M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    Stinespring W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)zbMATHMathSciNetGoogle Scholar
  67. 67.
    Tomamichel, M.: A Framework for Non-Asymptotic Quantum Information Theory. PhD thesis, ETH Zurich (2012) arXiv:1203.2142
  68. 68.
    Tomamichel, M.: Smooth entropies—a tutorial: with focus on applications in cryptography. Tutorial at QCRYPT 2012, slides available at, Sept (2012)
  69. 69.
    Unruh W.G.: Notes on black-hole evaporation. Phys. Rev. D 14(4), 870–892 (1976)ADSCrossRefGoogle Scholar
  70. 70.
    Wilde, M.M., Hayden, P., Buscemi, F., Hsieh, M.-H.: The information-theoretic costs of simulating quantum measurements. J. Phys. A Math. Theor. 45(45), 453001 (2012). arXiv:1206.4121
  71. 71.
    Winter A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inform. Theory 45(7), 2481–2485 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Winter, A.: Coding Theorems of Quantum Information Theory. PhD thesis, Universität Bielefeld (1999). arXiv:quant-ph/9907077
  73. 73.
    Winter, A.: “Extrinsic” and “intrinsic” data in quantum measurements: asymptotic convex decomposition of positive operator valued measures. Commun. Math. Phy. 244, 157 (2004). arXiv:quant-ph/0109050
  74. 74.
    Wolf, M.M.: Quantum channels & operations: Guided tour. Lecture Notes Available at July (2012)
  75. 75.
    Wolfowitz J.: The coding of messages subject to chance errors. Ill. J. Math. 1, 591–606 (1957)zbMATHMathSciNetGoogle Scholar
  76. 76.
    Wolfowitz J.: Coding Theorems of Information Theory. Prentice-Hall, Englewood Cliffs (1962)Google Scholar
  77. 77.
    Yard, J., Hayden, P., Devetak, I.: Capacity theorems for quantum multiple-access channels: classical-quantum and quantum-quantum capacity regions. IEEE Trans. Inform. Theory 54(7), 3091–3113 (2008). arXiv:quant-ph/0501045

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Physics and Astronomy, Center for Computation and Technology, Hearne Institute for Theoretical PhysicsLouisiana State UniversityBaton RougeUSA
  2. 2.ICREA, Informació i Fenomens, QuànticsUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  3. 3.Física Teòrica, Informació i Fenomens, QuànticsUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  4. 4.School of MathematicsUniversity of BristolBristolUK
  5. 5.Laboratory for Quantum InformationChina Jiliang UniversityHangzhouChina

Personalised recommendations