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Dividing Quantum Channels

Abstract

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

References

  1. Holevo, A.S.: Statistical Structure of Quantum Theory. Springer Lecture Notes in Physics, Berlin- Heidelberg-New York: Springer, 2001

  2. Horn R.A. (1967). Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8: 219

    MATH  Article  MathSciNet  Google Scholar 

  3. Holevo A.S. (1986). Theor. Probab. Appl. 32: 560

    Google Scholar 

  4. Denisov L.V. (1988). Th. Prob. Appl. 33: 392

    MATH  Article  MathSciNet  Google Scholar 

  5. Jamiolkowski A. (1972). Rep. Math. Phys. 3: 275

    MATH  Article  MathSciNet  Google Scholar 

  6. Choi M.D. (1975). Lin. Alg. Appl. 10: 285

    MATH  Article  Google Scholar 

  7. Kraus K. (1983). States, Effects and Operations. Springer, Berlin-Heidelberg-New York

    MATH  Google Scholar 

  8. Wolf M.M. and Perez-Garcia D. (2007). Phys. Rev. A 75: 012303

    Article  ADS  Google Scholar 

  9. Lindbald G. (1976). Commun. Math. Phys. 48: 119

    Article  ADS  Google Scholar 

  10. Gorini V., Kossakowski A. and Sudarshan E.C.G. (1976). J. Math. Phys. 17: 821

    Article  ADS  MathSciNet  Google Scholar 

  11. Davies E.B. (1980). Rep. Math. Phys. 17: 249

    MATH  Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Schmidt, W.M.: Diophantine Approximation. Lecture Notes in Math. 785, Berlin-Heidelberg-New York: Springer Verlag, 1980

  14. Bhatia, R.: Matrix Analysis. Springer Graduate Texts in Mathematics 169, Berlin-Heidelberg-New York: Springer, 1997

  15. Streater R.F. (1995). Statistical Dynamics. Imperial College Press, London

    MATH  Google Scholar 

  16. Wigner, E.P.: Gruppentheorie. Braunschweig: Vieweg 1931; Group Theory. London: Academic Press, 1959

  17. Bargmann V. (1964). J. Math. Phys. 5: 862

    MATH  Article  ADS  MathSciNet  Google Scholar 

  18. Kadison R. (1965). Topology 3(supp. 2): 177

    Article  MathSciNet  Google Scholar 

  19. Buscemi F., D’Ariano G.M., Keyl M., Perinotti P. and Werner R. (2005). J. Math. Phys. 46: 082109

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  21. Uhlmann A. (1976). Rep. Math. Phys. 9: 273

    MATH  Article  MathSciNet  Google Scholar 

  22. Stoermer E. (1963). Acta Math. 110: 233

    MATH  Article  MathSciNet  Google Scholar 

  23. King C. and Ruskai M.B. (2001). IEEE Trans. Info. Theory 47: 192

    MATH  Article  MathSciNet  Google Scholar 

  24. Fujiwara A. and Algoet P. (1999). Phys. Rev. A 59: 3290

    Article  ADS  Google Scholar 

  25. Ruskai M.B., Szarek S. and Werner E. (2002). Lin. Alg. Appl. 347: 159

    MATH  Article  MathSciNet  Google Scholar 

  26. Gorini V. and Sudarshan E.C.G. (1976). Commun. Math. Phys. 46: 43

    MATH  Article  ADS  MathSciNet  Google Scholar 

  27. Verstraete, F., Verschelde, H.: http://arxiv.org/list/quant-ph/0202124, 2002; F. Verstraete, J. Dehaene, B. De Moor.: Phys. Rev. A 64, 010101(R) (2001)

  28. Vollbrecht K.G.H. and Werner R.F. (2000). J. Math. Phys. 41: 6772

    MATH  Article  ADS  MathSciNet  Google Scholar 

  29. Bacon D., Childs A.M., Chuang I.L., Kempe J., Leung D.W. and Zhou X. (2001). Phys. Rev. A 64: 062302

    Article  ADS  Google Scholar 

  30. Eisert, J., Wolf, M.M.: http://arxiv.org/list/quant-ph/0505151, 2005; ‘Gaussian quantum channels’. In: Quantum Information with continuous variables of atoms and light, N. Cerf, G. Leuchs, E.S. Polzik (eds.) London: Imperial College Press, 2006

  31. Verstraete F., Cirac J.I., Latorre J.I., Rico E. and Wolf M.M. (2005). Phys. Rev. Lett. 94: 140601

    Article  ADS  Google Scholar 

  32. Wolf, M.M., Eisert, J., Cubitt, T.S., Cirac, J.I.: arXiv: 0711.3172 (2007)

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Correspondence to Michael M. Wolf.

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Communicated by M.B. Ruskai

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Wolf, M.M., Cirac, J.I. Dividing Quantum Channels. Commun. Math. Phys. 279, 147–168 (2008). https://doi.org/10.1007/s00220-008-0411-y

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Keywords

  • Quantum Channel
  • Positive Evolution
  • Trace Preserve
  • Markovian Approximation
  • Positive Operator Value Measure