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

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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.

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Beigi, S.: Sandwiched Rényi divergence satisfies data processing inequality. J. Math. Phys. 54(12), 122202 (2013). arXiv:1306.5920

  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. Berta, M., Renes, J.M., Wilde, M.M.: Identifying the information gain of a quantum measurement. (2013). arXiv:1301.1594

  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. 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. Brádler, K., Hayden, P., Panangaden, P.: Private information via the Unruh effect. J. High Energy Phys. 2009(08), 074 (2009). arXiv:0807.4536

  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)

  10. Buscemi, F., Hayashi, M., Horodecki, M.: Global information balance in quantum measurements. Phys. Rev. Lett. 100, 210504 (2008). arXiv:quant-ph/0702166

  11. Busch P.: Informationally complete sets of physical quantities. Int. J. Theor. Phys. 30(9), 1217–1227 (1991)

    Article  MathSciNet  Google Scholar 

  12. Carlen E.A.: Trace inequalities and quantum entropy: an introductory course. Contemp. Math. 529, 73–140 (2010)

    Article  MathSciNet  Google Scholar 

  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. 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. Csiszár I.: Generalized cutoff rates and Rényi’s information measures. IEEE Trans. Inform. Theory 41(1), 26–34 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. Dupuis, F., Fawzi, O., Wehner, S.: Entanglement sampling and applications. May 2013. arXiv:1305.1316

  20. Dupuis, F., Szehr, O., Tomamichel, M.: unpublished notes (2013)

  21. El Gamal A., Kim Y.-H.: Network Information Theory. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  22. Fehr, S.: On the conditional Rényi entropy. In: Lecture at the Beyond IID Workshop at the University of Cambridge, January (2013)

  23. Frank, R.L., Lieb, E.H.: Monotonicity of a relative Rényi entropy. J. Math. Phys. 54(12), 122201 (2013). arXiv:1306.5358

  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. Gallager R.G.: Information Theory and Reliable Communication. Wiley, New York (1968)

    MATH  Google Scholar 

  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. 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. Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255–257 (2009). arXiv:0809.3972

  29. Holevo A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inform. Transm. 9, 177–183 (1973)

    MathSciNet  Google Scholar 

  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. Holevo A.S.: Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1999)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  33. Holevo, A.S.: Information capacity of quantum observable. Probl. Inform. Transm. 48, 1 (2012) arXiv:1103.2615

  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. Jacobs, K.: On the properties of information gathering in quantum and classical measurements (2003). arXiv:quant-ph/0304200v1

  36. King, C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4653 (2002). arXiv:quant-ph/0103156

  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. King, C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inform. Theory 49(1), 221–229 (2003). arXiv:quant-ph/0204172

  39. King, C.: Maximal p-norms of entanglement breaking channels. Quantum Inform. Comput. 3(2), 186–190 (2003). arXiv:quant-ph/0212057

  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. Koenig, R., Wehner, S.: A strong converse for classical channel coding using entangled inputs. Phys. Rev. Lett. 103, 070504 (2009). arXiv:0903.2838

  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. Lamas-Linares A., Simon C., Howell J.C., Bouwmeester D.: Experimental quantum cloning of single photons. Science 296, 712–714 (2002)

    Article  ADS  Google Scholar 

  44. Leung, D., Smith, G.: Continuity of quantum channel capacities. Commun. Math. Phys. 292(1), 201–215 (2009). arXiv:0810.4931

  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)

  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. Matthews, W., Wehner, S.: Finite blocklength converse bounds for quantum channels. October (2012). arXiv:1210.4722

  48. Milonni P.W., Hardies M.L.: Photons cannot always be replicated. Phys. Lett. A 92(7), 321–322 (1982)

    Article  ADS  Google Scholar 

  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. Müller-Lennert, M.: Quantum relative Rényi entropies. Master’s thesis, ETH Zurich, April (2013)

  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. 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. Ohya M., Petz D., Watanabe N.: On capacities of quantum channels. Probab. Math. Stat. Wroclaw Univ. 17, 179–196 (1997)

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

  57. Prugovečki E.: Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 16, 321–331 (1977)

    Article  MATH  Google Scholar 

  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. Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)

    Article  ADS  Google Scholar 

  60. Schumacher B., Westmoreland M.D.: Optimal signal ensembles. Phys. Rev. A 63, 022308 (2001)

    Article  ADS  Google Scholar 

  61. Sharma, N., Warsi, N.A.: On the strong converses for the quantum channel capacity theorems (2012). arXiv:1205.1712

  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. Sibson R.: Information radius. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14(2), 149160 (1969)

    Article  MathSciNet  Google Scholar 

  64. Simon C., Weihs G., Zeilinger A.: Optimal quantum cloning via stimulated emission. Phys. Rev. Lett. 84(13), 2993–2996 (2000)

    Article  ADS  Google Scholar 

  65. Sion M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  66. Stinespring W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)

    MATH  MathSciNet  Google Scholar 

  67. Tomamichel, M.: A Framework for Non-Asymptotic Quantum Information Theory. PhD thesis, ETH Zurich (2012) arXiv:1203.2142

  68. Tomamichel, M.: Smooth entropies—a tutorial: with focus on applications in cryptography. Tutorial at QCRYPT 2012, slides available at, Sept (2012)

  69. Unruh W.G.: Notes on black-hole evaporation. Phys. Rev. D 14(4), 870–892 (1976)

    Article  ADS  Google Scholar 

  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. Winter A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inform. Theory 45(7), 2481–2485 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  72. Winter, A.: Coding Theorems of Quantum Information Theory. PhD thesis, Universität Bielefeld (1999). arXiv:quant-ph/9907077

  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. Wolf, M.M.: Quantum channels & operations: Guided tour. Lecture Notes Available at July (2012)

  75. Wolfowitz J.: The coding of messages subject to chance errors. Ill. J. Math. 1, 591–606 (1957)

    MATH  MathSciNet  Google Scholar 

  76. Wolfowitz J.: Coding Theorems of Information Theory. Prentice-Hall, Englewood Cliffs (1962)

    Google Scholar 

  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

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Correspondence to Mark M. Wilde.

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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, 593–622 (2014).

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