Some features of the conditional \({\mathsf q}\)-entropies of composite quantum systems

  • J. Batle
  • A. R. Plastino
  • M. Casas
  • A. PlastinoEmail author


The study of conditional q-entropies in composite quantum systems has recently been the focus of considerable interest, particularly in connection with the problem of separability. The q-entropies depend on the density matrix \(\rho\) through the quantity \(\omega_q = {\rm Tr}\rho^q\), and admit as a particular instance the standard von Neumann entropy in the limit case \(q\rightarrow 1\). A comprehensive numerical survey of the space of pure and mixed states of bipartite systems is here performed, in order to determine the volumes in state space occupied by those states exhibiting various special properties related to the signs of their conditional q-entropies and to their connections with other separability-related features, including the majorization condition. Different values of the entropic parameter q are considered, as well as different values of the dimensions N 1 and N 2 of the Hilbert spaces associated with the constituting subsystems. Special emphasis is paid to the analysis of the monotonicity properties, both as a function of q and as a function of N 1 and N 2, of the various entropic functionals considered.


Entropy Hilbert Space State Space Density Matrix Quantum System 
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  1. 1.
    R. Horodecki, P. Horodecki, M. Horodecki, Phys. Lett. A 210, 377 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Horodecki, M. Horodecki, Phys. Rev. A 54, 1838 (1996)CrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Cerf, C. Adami Phys. Rev. Lett. 79, 5194 (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    A. Vidiella-Barranco, Phys. Lett. A 260, 335 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    C. Tsallis, S. Lloyd, M. Baranger, Phys. Rev. A 63, 042104 (2001)CrossRefGoogle Scholar
  6. 6.
    C. Tsallis, P.W. Lamberti, D. Prato, Physica A 295, 158 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    F.C. Alcaraz, C. Tsallis, Phys. Lett. A 301, 105 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    C. Tsallis, D. Prato, C. Anteneodo, Eur. Phys. J. B 29, 605 (2002)CrossRefGoogle Scholar
  9. 9.
    B.M. Terhal, Theor. Comput. Sci. 287, 313 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    S. Abe, Phys. Rev. A 65, 052323 (2002)Google Scholar
  11. 11.
    K.G.B. Vollbrecht, M.M. Wolf, J. Math. Phys. 43, 4299 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    E. Schrödinger, Naturwissenschaften 23, 807 (1935)Google Scholar
  13. 13.
    Introduction to Quantum Computation and Information, edited by Hoi-Kwong Lo, S. Popescu, T. Spiller (World Scientific, River Edge, 1998)Google Scholar
  14. 14.
    C.P. Williams, S.H. Clearwater, Explorations in Quantum Computing (Springer, New York, 1997)Google Scholar
  15. 15.
    Quantum Computing and Quantum Communications, edited by C.P. Williams (Springer, Berlin, 1998)Google Scholar
  16. 16.
    The Physics of Quantum Information, edited by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer, Berlin, Heidelberg, 1998)Google Scholar
  17. 17.
    G. Alber, T. Beth, P. Horodecki, R. Horodecki, M. Röttler, H. Weinfurter, R. Werner, A. Zeilinger, Quantum Information, Springer Tracts in Modern Physics, Vol. 173 (Springer, Berlin, 2001)Google Scholar
  18. 18.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)Google Scholar
  19. 19.
    A. Galindo, M.A. Mart\’i n-Delgado, Rev. Mod. Phys. 74, 347 (2002)CrossRefGoogle Scholar
  20. 20.
    C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1993)zbMATHGoogle Scholar
  22. 22.
    A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993)Google Scholar
  23. 23.
    K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstein, Phys. Rev. A 58, 883 (1998)CrossRefMathSciNetGoogle Scholar
  24. 24.
    K. Zyczkowski, Phys. Rev. A 60, 3496 (1999)MathSciNetGoogle Scholar
  25. 25.
    K. Zyczkowski, H.J. Sommers, J. Phys. A 34, 7111 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    W.J. Munro, D.F.V. James, A.G. White, P.G. Kwiat, Phys. Rev. A 64, 030302 (2001)Google Scholar
  27. 27.
    S. Ishizaka, T. Hiroshima, Phys. Rev. A 62, 022310 (2000)Google Scholar
  28. 28.
    J. Batle, M. Casas, A.R. Plastino, A. Plastino Phys. Lett. A 298, 301 (2002)zbMATHGoogle Scholar
  29. 29.
    J. Batle, M. Casas, A.R. Plastino, A. Plastino, Phys. Lett. A 296, 251 (2002)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J. Batle, A.R. Plastino, M. Casas, A. Plastino, Phys. Lett. A 307, 253 (2003)zbMATHGoogle Scholar
  31. 31.
    M.A. Nielsen, J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)Google Scholar
  32. 32.
    C. Beck, F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993)Google Scholar
  33. 33.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988)MathSciNetzbMATHGoogle Scholar
  34. 34.
    P.T. Landsberg, V. Vedral, Phys. Lett. A 247, 211 (1998)MathSciNetzbMATHGoogle Scholar
  35. 35.
    J.A.S. Lima, R. Silva, A.R. Plastino, Phys. Rev. Lett. 86, 2938 (2001)CrossRefGoogle Scholar
  36. 36.
    A. Peres, Phys. Rev. Lett. 77, 1413 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996)MathSciNetzbMATHGoogle Scholar
  38. 38.
    J. Batle, A.R. Plastino, C. Casas, A. Plastino, J. Phys A: Math. Gen. 35, 10311 (2002)CrossRefGoogle Scholar
  39. 39.
    M. Pozniak, K. Zyczkowski, M. Kus, J. Phys A: Math. Gen. 31, 1059 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  • J. Batle
    • 1
  • A. R. Plastino
    • 1
    • 2
    • 3
  • M. Casas
    • 1
  • A. Plastino
    • 2
    Email author
  1. 1.Departament de Fí sicaUniversitat de les Illes BalearsPalma de MallorcaSpain
  2. 2.Physics DepartmentUniversity of PretoriaPretoria 0002South Africa
  3. 3.National University La Plata and CONICETLa PlataArgentina

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