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Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem

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  • Published: 26 May 2009
  • volume 289, pages 1057–1086 (2009)
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Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem
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  • Christian B. Mendl1 &
  • Michael M. Wolf1,2 
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Abstract

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel. These scenarios imply that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.

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References

  1. Pérez-García D., Wolf M.M., Petz D., Ruskai M.B.: Contractivity of positive and trace-preserving maps under L p norms. J. Math. Phys. 47(8), 083506 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  2. Arias A., Gheondea A., Gudder S.: Fixed points of quantum operations. J. Math. Phys. 43(12), 5872–5881 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. King C.: Additivity for unital qubit channels. J. Math. Phys. 43, 4641–4653 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Fukuda M.: Simplification of additivity conjecture in quantum information theory. Quant. Inf. Comp. 6, 179–186 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fukuda M., Wolf M.M.: Simplifying additivity problems using direct sum constructions. J. Math. Phys. 48, 2101 (2007)

    Article  MathSciNet  Google Scholar 

  6. Rosgen B.: Additivity and distinguishability of random unitary channels. J. Math. Phys. 49, 102107 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  7. Tregub S.L.: Bistochastic operators on finite-dimensional von-Neumann algebras. Soviet Math. 30, 105 (1986)

    MATH  Google Scholar 

  8. Kümmerer B., Maassen H.: The essentially commutative dilations of dynamical semigroups on M n . Commun. Math. Phys. 109, 1–22 (1987)

    Article  MATH  ADS  Google Scholar 

  9. Landau L.J., Streater R.F.: On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras. Lin. Alg. Appl. 193, 107–127 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Birkhoff G.: Three observations on linear algebra. Univ. Nac. Tucuan. Revista A 5, 147–151 (1946)

    MATH  MathSciNet  Google Scholar 

  11. Gregoratti M., Werner R.F.: Quantum lost and found. J. Mod. Opt. 50, 915–933 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  12. Smolin J.A., Verstraete F., Winter A.: Entanglement of assistance and multipartite state distillation. Phys. Rev. A 72, 052317 (2005)

    Article  ADS  Google Scholar 

  13. Jamiolkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Watrous, J.: Mixing doubly stochastic quantum channels with the completely depolarizing channel. http://arXiv.org/abs/0807.2668v1[quant-ph], 2008

  15. Rudolph O.: On extremal quantum states of composite systems with fixed marginals. J. Math. Phys. 45(11), 4035–4041 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Parthasarath K.R.: Extremal quantum states in coupled systems. Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257–268 (2005)

    Article  ADS  Google Scholar 

  17. Audenaert K.M.R., Scheel S.: On random unitary channels. New J. Phys. 10, 023011 (2008)

    Article  ADS  Google Scholar 

  18. Choi M.-D.: Completely positive linear maps on complex matrices. Lin. Alg. Appl. 10, 285–290 (1975)

    Article  MATH  Google Scholar 

  19. Buscemi F.: On the minimal number of unitaries needed to describe a random-unitary channel. Phys. Lett. A 360, 256–258 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Zeidler E.: Applied Functional Analysis. Main Principles and Their Applications. Springer, New York (1995)

    MATH  Google Scholar 

  21. DiVincenzo, D.P., Fuchs, C.A., Mabuchi, H., Smolin, J.A., Thapliyal, A.V., Uhlmann, A.: Entanglement of assistance. In: Proc. Quantum Computing and Quantum Communications, First NASA Intl. Conf., Palm Springs, Berlin-Heidelberg-New York: Springer, 1999, pp. 247–257

  22. Vidal G., Werner R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    Article  ADS  Google Scholar 

  23. Werner R.F., Holevo A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353–4357 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Vollbrecht K.G.H., Werner R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)

    Article  ADS  Google Scholar 

  25. Faßbender H., Ikramov Kh.D.: Some observations on the Youla form and conjugate-normal matrices. Lin. Alg. Appl. 422, 29–38 (2006)

    Article  Google Scholar 

  26. Open problems in QIT. http://www.imaph.tu-bs.de/qi/problems/

  27. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge, Cambridge University Press, 1990

    MATH  Google Scholar 

  28. Bunse-Gerstner A., Byers R., Mehrmann V.: A quaternion qr algorithm. Numer. Math. 55, 83–95 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institute for Quantum Optics, Garching, Germany

    Christian B. Mendl & Michael M. Wolf

  2. Niels Bohr Institute, Copenhagen, Denmark

    Michael M. Wolf

Authors
  1. Christian B. Mendl
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  2. Michael M. Wolf
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Corresponding author

Correspondence to Michael M. Wolf.

Additional information

Communicated by M.B. Ruskai

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Mendl, C.B., Wolf, M.M. Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem. Commun. Math. Phys. 289, 1057–1086 (2009). https://doi.org/10.1007/s00220-009-0824-2

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  • Received: 13 July 2008

  • Accepted: 05 March 2009

  • Published: 26 May 2009

  • Issue Date: August 2009

  • DOI: https://doi.org/10.1007/s00220-009-0824-2

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Keywords

  • Convex Hull
  • Extreme Point
  • Entangle State
  • Quantum Channel
  • Convex Combination

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