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Topics in Operator Theory pp 535–538Cite as

On Norms of Completely Positive Maps

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 202))

Abstract

King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained.

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References

  1. C. King and M.B. Ruskai, Comments on multiplicativity of maximal p-norms when p = 2, Quantum Inf. Comput. 4 (2004), no. 6-7, 500–512.

    MATH  MathSciNet  Google Scholar 

  2. J. Watrous, Notes on super-operator norms induced by Schatten norms, Quantum Inf. Comput. 5 (2005), 58–68.

    MathSciNet  Google Scholar 

  3. K. Audenaert, A Note on the p → q norms of Completely Positive Maps, arxiv.org e-print math-ph/0505085. [A note on the p → q norms of 2-positive maps. Linear Algebra Appl. 430 (2009), no. 4, 1436–1440].

    Article  MATH  MathSciNet  Google Scholar 

  4. A.S. Kholevo, Multiplicativity of p-norms of completely positive maps and the additivity problem in quantum information theory, Russian Math. Surveys 61 (2006), no. 2, 301–339.

    Article  MathSciNet  Google Scholar 

  5. I. Devetak, M. Junge, C. King and M.B. Ruskai, Multiplicativity of completely bounded p-norms implies a new additivity result, Commun. Math. Phys. 266 (2006), 37–63.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Hayden and A. Winter, Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys. 284 (2008), no. 1, 263–280.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Schatten, Norm Ideals of Completely Continuous Operators. Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  8. R.A. Horn and C.R. Johnson, Topics in matrix analysis. Cambridge University Press, Cambridge, 1994.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Université Pierre et Marie Curie-Paris 6, UMR 7586-Institut de Mathématiques, BC 186, F-75252, Paris, France

    Stanislaw J. Szarek

  2. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, 44106-7058, USA

    Stanislaw J. Szarek

Authors
  1. Stanislaw J. Szarek
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Additional information

Communicated by L. Rodman.

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© 2010 Birkhäuser Verlag Basel/Switzerland

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Szarek, S.J. (2010). On Norms of Completely Positive Maps. In: Topics in Operator Theory. Operator Theory: Advances and Applications, vol 202. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0158-0_31

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