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Spectral Conditions for Positive Maps

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Abstract

We provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes the celebrated Choi example of a map which is positive but not completely positive. It is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements. Such maps provide basic tools to study quantum entanglement in multipartite systems.

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References

  1. Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Peres A.: Phys. Rev. Lett. 77, 1413 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Horodecki P.: Phys. Lett. A 232, 333 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Størmer E.: Acta Math. 110, 233 (1963)

    Article  MathSciNet  Google Scholar 

  5. Arveson W.: Acta Math. 123, 141 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  6. Choi, M.-D.: Lin. Alg. Appl. 10, 285 (1975); ibid 12, 95 (1975)

  7. Woronowicz S.L.: Rep. Math. Phys. 10, 165 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Woronowicz S.L.: Commun. Math. Phys. 51, 243 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Perez-Garcia D., Wolf M.M., Petz D., Ruskai M.B.: J. Math. Phys. 47, 083506 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  10. Chruściński D., Kossakowski A.: J. Phys. A: Math. Theor. 41, 215201 (2008)

    Article  ADS  Google Scholar 

  11. Størmer, E.: J. Funct. Anal. 254, 2303 (2008); E. Størmer: Separable states and positive maps II. http://arxiv.org/abs/:0803.4417V1[math.OA], 2008

  12. Chruściński D., Kossakowski A.: Open Syst. and Inf. Dyn. 14, 275 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Paulsen V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  14. Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)

    MATH  Google Scholar 

  15. Terhal B., Horodecki P.: Phys. Rev. A 61, 040301 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  16. Jamiołkowski A.: Rep. Math. Phys. 3, 275 (1972)

    Article  MATH  ADS  Google Scholar 

  17. Kraus K.: States, Effects and Operations: Fundamental Notions of Quantum Theory. Berlin-Heidelberg Springer Verlag, NewYork (1983)

    Book  MATH  Google Scholar 

  18. Takasaki K., Tomiyama J.: Math. Zeit. 184, 101–108 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Benatti F., Floreanini R., Piani M.: Open Syst. and Inf. Dyn. 11, 325–338 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Horodecki M., Horodecki P., Horodecki R.: Phys. Lett. A 283, 1 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Chruściński, D., Kossakowski, A.: Phys. Rev. A 73, 062313 (2006); Phys. Rev. A 73:062314 (2006)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Physics, Nicolaus Copernicus University, Grudzia̧dzka 5/7, 87–100, Toruń, Poland

    Dariusz Chruściński & Andrzej Kossakowski

Authors
  1. Dariusz Chruściński
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  2. Andrzej Kossakowski
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Correspondence to Dariusz Chruściński.

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Communicated by M. B. Ruskai

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Chruściński, D., Kossakowski, A. Spectral Conditions for Positive Maps. Commun. Math. Phys. 290, 1051–1064 (2009). https://doi.org/10.1007/s00220-009-0790-8

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  • DOI: https://doi.org/10.1007/s00220-009-0790-8

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