Communications in Mathematical Physics

, Volume 297, Issue 2, pp 345–370 | Cite as

Random Quantum Channels I: Graphical Calculus and the Bell State Phenomenon

Article

Abstract

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel.

As an application, we study variations of random matrix models introduced by Hayden [7], and show that their eigenvalues converge almost surely.

In particular we obtain, for some models, sharp improvements on the value of the largest eigenvalue, and this is shown in further work to have new applications to minimal output entropy inequalities.

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References

  1. 1.
    Coecke, B.: Kindergarten quantum mechanics—lecture notes. In: Quantum Theory: Reconsideration of Foundations—3, AIP Conf. Proc. 810, Melville, NY: Amer. Inst. Phys., 2006, pp. 81–98Google Scholar
  2. 2.
    Collins B.: Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. Int. Math. Res. Not. 17, 953–982 (2003)CrossRefGoogle Scholar
  3. 3.
    Collins, B., Nechita, I.: Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems. http://arxiv.org/abs/0906.1877v2[math.PR], 2009
  4. 4.
    Collins, B., Nechita, I.: Gaussianization and eigenvalue statistics for Random quantum channels (III) http://arxiv.org/abs/0910.1768v2[quant-ph], 2009
  5. 5.
    Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)MATHCrossRefADSGoogle Scholar
  6. 6.
    Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nature Physics 5, 255–257 (2009)CrossRefADSGoogle Scholar
  7. 7.
    Hayden, P.: The maximal p-norm multiplicativity conjecture is false. http://arxiv.org/abs/0707.3291v1[quant-ph], 2007
  8. 8.
    Hayden P., Leung D., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Hayden P., Winter A.: Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1. Commun. Math. Phys. 284(1), 263–280 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Jones, V.F.R.: Planar Algebras. http://arxiv.org/abs/math/9909027v1[math.QA], 1999
  11. 11.
    Joyal A., Street R., Verity D.: Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119(3), 447–468 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Nechita I.: Asymptotics of random density matrices. Ann. Henri Poincaré 8(8), 1521–1538 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Volume 335 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 2006Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Département de Mathématique et StatistiqueUniversité d’OttawaOttawaCanada
  2. 2.Université de Lyon, Institut Camille JordanVilleurbanne CedexFrance

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