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On Quantum Estimation, Quantum Cloning and Finite Quantum de Finetti Theorems

  • Giulio Chiribella
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6519)

Abstract

This paper presents a series of results on the interplay between quantum estimation, cloning and finite de Finetti theorems. First, we consider the measure-and-prepare channel that uses optimal estimation to convert M copies into k approximate copies of an unknown pure state and we show that this channel is equal to a random loss of all but s particles followed by cloning from s to k copies. When the number k of output copies is large with respect to the number M of input copies the measure-and-prepare channel converges in diamond norm to the optimal universal cloning. In the opposite case, when M is large compared to k, the estimation becomes almost perfect and the measure-and-prepare channel converges in diamond norm to the partial trace over all but k systems. This result is then used to derive de Finetti-type results for quantum states and for symmetric broadcast channels, that is, channels that distribute quantum information to many receivers in a permutationally invariant fashion. Applications of the finite de Finetti theorem for symmetric broadcast channels include the derivation of diamond-norm bounds on the asymptotic convergence of quantum cloning to state estimation and the derivation of bounds on the amount of quantum information that can be jointly decoded by a group of k receivers at the output of a symmetric broadcast channel.

Keywords

Broadcast Channel Partial Trace Quantum Capacity Quantum Cloning Random Loss 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Gisin, N., Massar, S.: Phys. Rev. Lett. 79, 2153 (1997)CrossRefGoogle Scholar
  2. 2.
    Werner, R.F.: Phys. Rev. A 58, 1827 (1998)CrossRefGoogle Scholar
  3. 3.
    Bruß, D., Ekert, A., Macchiavello, C.: Phys. Rev. Lett. 81, 2598 (1998)CrossRefGoogle Scholar
  4. 4.
    Keyl, M., Werner, R.F.: J. Math. Phys. 40, 3283 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bruß, D., Cinchetti, M., DAriano, G.M., Macchiavello, C.: Phys. Rev. A 62, 12302 (2000)CrossRefGoogle Scholar
  6. 6.
    D’Ariano, G.M., Macchiavello, C.: Phys. Rev. A 67, 042306 (2003)CrossRefGoogle Scholar
  7. 7.
    Bae, J., Acín, A.: Phys. Rev. Lett. 97, 30402 (2006)CrossRefGoogle Scholar
  8. 8.
    Chiribella, G., D’Ariano, G.M.: Phys. Rev. Lett. 97, 250503 (2006)CrossRefGoogle Scholar
  9. 9.
    Keyl, M.: Problem 22 of the list, http://www.imaph.tu-bs.de/qi/problems/
  10. 10.
    Christandl, M., Koenig, R., Mitchison, G., Renner, R.: Comm. Math. Phys. 273, 473 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Renner, R.: Nature Physics 3, 645 (2007)CrossRefGoogle Scholar
  12. 12.
    Koenig, R., Mitchison, G.: J. Math. Phys. 50, 12105 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Caves, C.M., Fuchs, C.A., Schack, R.: J. Math. Phys. 43, 4537 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    de Finetti, B.: Theory of Probability. Wiley, New York (1990)MATHGoogle Scholar
  15. 15.
    Massar, S., Popescu, S.: Phys. Rev. Lett. 74, 1259 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Scarani, V., Iblisdir, S., Gisin, N., Acín, A.: Rev. Mod. Phys. 77, 1225 (2005)CrossRefGoogle Scholar
  17. 17.
    Aharonov, D., Kitaev, A., Nisan, N.: Quantum Circuits with Mixed States. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC). ACM, New York (1998)Google Scholar
  18. 18.
    Paulsen, V.I.: Completely bounded maps and dilations. Longman Scientific and Technical (1986)Google Scholar
  19. 19.
    Askey, R.: Orthogonal polynomials and special functions, Philadelphia, PA. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 21 (1975)Google Scholar
  20. 20.
    Yard, J., Hayden, P., Devetak, I.: arXiv:quant-ph/0603098v1Google Scholar
  21. 21.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: J. Math. Phys. 50, 42101 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Leung, D., Smith, G.: Comm. Math. Phys. 292, 201 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Holevo, A.S., Werner, R.F.: Phys. Rev. A 3, 32312 (2001)CrossRefGoogle Scholar
  24. 24.
    Klee, V.: Canad. J. Math. 16, 517 (1963)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Giulio Chiribella
    • 1
  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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