Communications in Mathematical Physics

, Volume 305, Issue 1, pp 85–97

Hastings’s Additivity Counterexample via Dvoretzky’s Theorem

  • Guillaume Aubrun
  • Stanisław Szarek
  • Elisabeth Werner


The goal of this note is to show that Hastings’s counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky’s theorem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  2. 2.
    Holevo, A.S.: The additivity problem in quantum information theory. In: “Proceedings of the International Congress of Mathematicians (Madrid, 2006),” Vol. III, Zürich: Eur. Math. Soc., 2006, pp. 999–1018Google Scholar
  3. 3.
    Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nature Phys. 5, 255 (2009)ADSCrossRefGoogle Scholar
  5. 5.
    Dvoretzky, A.: Some Results on Convex Bodies and Banach Spaces. In: “Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960),” Jerusalem: Jerusalem Academic Press, Oxford: Pergamon, 1961, pp. 123–160Google Scholar
  6. 6.
    Brandao F., Horodecki M.: On Hastings’ counterexamples to the minimum output entropy additivity conjecture. Open Syst. Inf. Dyn. 17, 31 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fukuda M., King C., Moser D.: Comments on Hastings’ Additivity Counterexamples. Commun. Math. Phys. 296, 111 (2010)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Hayden P., Winter A.: Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1. Commun. Math. Phys. 284, 263–280 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Aubrun G., Szarek S., Werner E.: Non-additivity of Rényi entropy and Dvoretzky’s theorem. J. Math. Phys. 51, 022102 (2010)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Milman V.: A new proof of the theorem of A. Dvoretzky on sections of convex bodies. Funct. Anal. Appl. 5, 28–37 (1971) (English translation)MathSciNetGoogle Scholar
  11. 11.
    Figiel T., Lindenstrauss J., Milman V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139(1-2), 53–94 (1977)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Collins, B., Nechita, I.: Gaussianization and eigenvalue statistics for random quantum channels (III), Ann. Appl. Probab., to appear; [quant-ph], 2009
  13. 13.
    Lévy P.: Problémes concrets d’analyse fonctionnelle. 2nd ed. Gauthier-Villars, Paris (1951)MATHGoogle Scholar
  14. 14.
    Pisier, G.: The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge: Cambridge University Press, 1989Google Scholar
  15. 15.
    Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in R n. In: “Geometric aspects of functional analysis (1986/87),” Lecture Notes in Math., 1317 Berlin: Springer, 1988, pp. 84–106Google Scholar
  16. 16.
    Schechtman, G.: A remark concerning the dependence on \({\varepsilon}\) in Dvoretzky’s theorem. In: “Geometric aspects of functional analysis (1987–88),” Lecture Notes in Math., 1376 Berlin: Springer, 1989, pp. 274–277Google Scholar
  17. 17.
    Marchenko V.A., Pastur L.A.: The distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72, 507–536 (1967)MathSciNetGoogle Scholar
  18. 18.
    Hayden P., Leung D., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Haagerup U., Thorbjørnsen S.: Random matrices with complex Gaussian entries. Expos. Math. 21, 293–337 (2003)MATHCrossRefGoogle Scholar
  20. 20.
    Geman S.: A limit theorem for the norm of random matrices. Ann. Probab. 8, 252–261 (1980)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Silverstein J.W.: The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab 13, 1364–1368 (1985)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Dudley R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Jain, N. C., Marcus, M. B.: Continuity of subgaussian processes. In: “Probability on Banach Spaces,” Advances in Probability, Vol. 4, New York: Dekker, 1978, pp. 81–196Google Scholar
  24. 24.
    Talagrand M.: The generic chaining. Upper and Lower bounds of Stochastic Processes. Springer, Berlin-Heidelberg-New York (2005)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Guillaume Aubrun
    • 1
  • Stanisław Szarek
    • 2
    • 3
  • Elisabeth Werner
    • 3
    • 4
  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance
  2. 2.Equipe d’Analyse Fonctionnelle, Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie-Paris 6ParisFrance
  3. 3.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  4. 4.Université de Lille 1, UFR de MathématiqueVilleneuve d’AscqFrance

Personalised recommendations