Communications in Mathematical Physics

, Volume 284, Issue 1, pp 263–280 | Cite as

Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

Article

Abstract

For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.

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References

  1. 1.
    Schumacher B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Jozsa R., Schumacher B.: A new proof of the quantum noiseless coding theorem. J. Mod. Opt. 41, 2343–2349 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Pierce J.: The early days of information theory. IEEE Transactions on Information Theory 19(1), 3–8 (1973)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gordon, J.P.: Noise at optical frequencies; information theory. In: Miles P.A. ed., Quantum electronics and coherent light; Proceedings of the international school of physics Enrico Fermi, Course XXXI, New York: Academic Press, 1964 pp. 156–181Google Scholar
  5. 5.
    Holevo, A.S.: Information theoretical aspects of quantum measurements. Probl. Info. Transm. (USSR), 9(2), 31–42 (1973). Translation: Probl. Info. Transm. 9, 177–183 (1973)Google Scholar
  6. 6.
    Hausladen P., Jozsa R., Schumacher B., Westmoreland M., Wootters W.K.: Classical information capacity of a quantum channel. Phys. Rev. A 54, 1869–1876 (1996)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)CrossRefADSGoogle Scholar
  9. 9.
    Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, 453–472 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Pomeransky A.A.: Strong superadditivity of the entanglement of formation follows from its additivity. Physical Review A 68(3), 032317 (2003)CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Audenaert K.M.R., Braunstein S.L.: On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246, 443–452 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Matsumoto K., Shimono T., Winter A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. Commun. Math. Phys. 246, 427–442 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Bennett C.H., DiVincenzo D.P., Smolin J.A., Wootters W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Hayden P.M., Horodecki M., Terhal B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen. 34, 6891–6898 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Vidal G., Dür W., Cirac J.I.: Entanglement cost of bipartite mixed states. Phys. Rev. Lett. 89(2), 027901 (2002)CrossRefADSGoogle Scholar
  16. 16.
    Matsumoto K., Yura F.: Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels. J. Phys. A: Math. Gen. 37, L167–L171 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Vollbrecht K.G.H., Werner R.F.: Entanglement measures under symmetry. Phys. Rev. A 64(6), 062307 (2001)CrossRefADSGoogle Scholar
  18. 18.
    King C., Ruskai M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Inf. Th. 47(1), 192–209 (2001)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Osawa S., Nagaoka H.: Numerical experiments on the capacity of quantum channel with entangled input states. IEICE Trans. Fund. Elect., Commun. and Comp. Sci. E84(A10), 2583–2590 (2001)Google Scholar
  20. 20.
    Amosov G.G., Holevo A.S., Werner R.F.: On some additivity problems of quantum information theory. Probl. Inform. Transm. 36(4), 25 (2000)Google Scholar
  21. 21.
    Amosov, G.G., Holevo, A.S.: On the multiplicativity conjecture for quantum channels. http://arxiv.org/list/:math-ph/0103015, 2001
  22. 22.
    King C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4643 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Fujiwara A., Hashizumé T.: Additivity of the capacity of depolarizing channels. Phys. Lett. A 299, 469–475 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    King C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inf. Th. 49(1), 221–229 (2003)MATHCrossRefGoogle Scholar
  25. 25.
    Holevo A.S.: Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1998)MATHCrossRefGoogle Scholar
  26. 26.
    King, C.: Maximization of capacity and p-norms for some product channels. http://arxiv.org/list/:quant-ph/0103086, 2001
  27. 27.
    Shor P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43, 4334–4340 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    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)MATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    King, C., Matsumoto, K., Nathanson, M., Ruskai, M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. http://arxiv.org/list/:quant-ph/0509126, 2005., to appear in special issue of Markov processes and Related Fields in memory of J.F. leuis
  30. 30.
    Cortese J.: Holevo-Schumacher-Westmoreland channel capacity for a class of qudit unital channels. Phys. Rev. A 69(2), 022302 (2004)CrossRefADSGoogle Scholar
  31. 31.
    Datta, N., Holevo, A.S., Suhov, Y.M.: A quantum channel with additive minimum output entropy. http://arxiv.org/list/:quant-ph/0403072, 2004
  32. 32.
    Fukuda M.: Extending additivity from symmetric to asymmetric channels. J. Phys. A: Math. Gen. 38, L753–L758 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Holevo, A.S.: Additivity of classical capacity and related problems. Available online at: http://www.imaph.tu-bs.de/qi/problems/10.pdf, 2004
  34. 34.
    Holevo, A.S.: The additivity problem in quantum information theory. In: Proceedings of the International Congress of Mathematicians, (Madrid, Spain, 2006), Zurich:Publ. EMS, 2007, pp. 999–1018Google Scholar
  35. 35.
    King C., Ruskai M.B.: Comments on multiplicativity of maximal p-norms when p = 2. Quantum Inf. and Comput. 4, 500–512 (2004)MATHMathSciNetGoogle Scholar
  36. 36.
    King C., Nathanson M., Ruskai M.B.: Multiplicativity properties of entrywise positive maps. Linear alge. Applications. 404, 367–379 (2005)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Serafini A., Eisert J., Wolf M.M.: Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs. Phys. Rev. A 71(1), 012320 (2005)CrossRefADSGoogle Scholar
  38. 38.
    Giovannetti V., Lloyd S.: Additivity properties of a Gaussian channel. Phys. Rev. A 69, 062307 (2004)CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Devetak I., Junge M., King C., Ruskai M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37–63 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Michalakis, S.: Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels. http://arxiv.org/list/:0707.1722, 2007
  41. 41.
    Werner R.F., Holevo A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353–4357 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Alicki R., Fannes M.: Note on multiple additivity of minimal Renyi entropy output of the Werner-Holevo channels. Open Syst. Inf. Dyn. 11(4), 339–342 (2005)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Datta, N.: Multiplicativity of maximal p-norms in Werner-Holevo channels for 1 < p < 2. http://arxiv.org/list/:quant-ph/0410063, 2004
  44. 44.
    Giovannetti V., Lloyd S., Ruskai M.B.: Conditions for multiplicativity of maximal p -norms of channels for fixed integer p. J. Math. Phys. 46, 042105 (2005)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Winter, A.: The maximum output p-norm of quantum channels is not multiplicative for any p > 2. http://arxiv.org/abs/:0707.0402, 2007
  46. 46.
    Hayden, P.: The maximal p-norm multiplicativity conjecture is false. arXiv.org:0707.3291, 2007
  47. 47.
    Hayden P., Leung D., Shor P.W., Winter A.: Randomizing Quantum States: Constructions and Applications. Commun. Math. Phys. 250, 371–391 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Aubrun, G.: On almost randomizing channels with a short Kraus decomposition. http://arxiv.org/abs/:0805.2900v2, 2008
  49. 49.
    Paulsen, V.I.: Completely bounded maps and dilations. New York: Longman Scientific and Technical, 1986Google Scholar
  50. 50.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Bennett C.H., Hayden P., Leung D.W., Shor P.W., Winter A.: Remote preparation of quantum states. IEEE Trans. Inf. Th. 51(1), 56–74 (2005)CrossRefMathSciNetGoogle Scholar
  52. 52.
    Geman S.: A Limit Theorem for the Norm of Random Matrices. Ann. Prob. 8(2), 252–261 (1980)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Johnstone I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29(2), 295–327 (2001)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Davidson, K.R., Szarek, S.J.: Local Operator Theory, Random Matrices and Banach Spaces. In: Johnson W.B., Lindenstrauss J. eds., Handbook of the Geometry of Banach Spaces, Vol. I, Chap. 8, London:Elsevier, 2001, pp. 317–366Google Scholar
  55. 55.
    Ledoux, M.: The concentration of measure phenomenon, Vol. 89 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 2001Google Scholar
  56. 56.
    King C.: Maximal p-norms of entanglement breaking channels. Quantum Inf. and Comp. 3(2), 186–190 (2003)MATHGoogle Scholar
  57. 57.
    Wolf M.M., Eisert J.: Classical information capacity of a class of quantum channels. New J. Phys. 7, 93 (2005)CrossRefADSGoogle Scholar
  58. 58.
    Cubitt, T., Harrow, A.W., Leung, D., Montanaro, A., Winter, A.: Counterexamples to additivity of minimum output p-Rényi entropy for p close to 0. http://arxiv.org/abs/:0712.3628v2, 2007, Commun. Math. Phys. doi:10.1007/s00220-008-0625-z
  59. 59.
    Ambainis, A., Smith, A.: Small pseudo-random families of matrices: Derandomizing approximate quantum encryption. In: Proc. RANDOM, LNCS 3122, Berlin-Heidelberg-NewYork: Springer, 2004, pp. 249–260Google Scholar
  60. 60.
    Ben-Aroya, A., Ta-Shma, A.: Quantum expanders and the quantum entropy difference problem. http://arxiv.org/abs/:quant-ph/0702129, 2007
  61. 61.
    Hastings M.B.: Random unitaries give quantum expanders. Phys. Rev. A 76, 032315 (2007)CrossRefADSGoogle Scholar
  62. 62.
    Pérez-García D., Wolf M.M., Palazuelos C., Villanueva I., Junge M.: Unbounded Violation of Tripartite Bell Inequalities. Commun. Math. Phys. 279(2), 455–486 (2008)CrossRefADSGoogle Scholar
  63. 63.
    Aubert S., Lam C.S.: Invariant integration over the unitary group. J. Math. Phys. 44, 6112–6131 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  64. 64.
    Aubert S., Lam C.S.: Invariant and group theoretical integrations over the U(n) group. J. Math. Phys. 45, 3019–3039 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  65. 65.
    Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006)MATHCrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Department of MathematicsUniversity of BristolBristolUK
  3. 3.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore

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