Open Systems & Information Dynamics

, Volume 5, Issue 3, pp 209–228 | Cite as

Entropy and Optimal Decompositions of States Relative to a Maximal Commutative Subalgebra

  • Armin Uhlmann


To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor.

I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis is collected in an appendix.


Entropy Statistical Physic Mechanical Engineer System Theory General Property 
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  1. 1.
    R. T. Rockafellar, Conver Analysis, Princeton University Press, 1970.Google Scholar
  2. 2.
    A. S. Holevo, Probl. Peredachi Inform 8, 63 (1972); Probl. Peredachi Inform 9, 3 (1973); Rep. Math. Phys. 12, 273 (1977); P. Hausladen, R. Josza, B. Schumacher, M. Westmoreland, W. Wootters, Phys. Rev. A 54, 1896 (1996); A. S. Holevo, quant-ph/9611023; Christopher Fuchs. private communication.Google Scholar
  3. 3.
    A. Wehrl, Rev. Mod. Phys. 50, 221 (1978).Google Scholar
  4. 4.
    E. D. Davies, IEEE Trans. Inf. Theory IT-24, 596 (1978).Google Scholar
  5. 5.
    H. Narnhofer, W. Thirring, Fizika 17, 257 (1985).Google Scholar
  6. 6.
    A. Connes, C. R. Acad. Sci. Paris 301 I, 1 (1985)Google Scholar
  7. 7.
    A. Connes, H. Narnhofer, W. Thirring, Comm. Math. Phys. 112, 681 (1987).Google Scholar
  8. 8.
    M. Ohya, D. Petz, Quantum Entropy and Its Use, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993.Google Scholar
  9. 9.
    L. B. Levitin, Open Sys. Information Dyn., 2, 319 (1994)Google Scholar
  10. 10.
    F. Benatti, H. Narnhofer, A. Uhlmann, Rep. Math. Phys. 38, 123 (1996).Google Scholar
  11. 11.
    F. Benatti, J. Math. Phys 37, 5244 (1996).Google Scholar
  12. 12.
    Ch.A. Fuchs, A. Peres, Phys. Rev. A 53, 2038 (1996).Google Scholar
  13. 13.
    C. H. Bennett, D. V. DiVincenzo, J. Smolin, W. K. Wootters, Phys. Rev. A 54, 3824 (1996), quant-ph/9604024 v2.Google Scholar
  14. 14.
    A. Uhlmann, Optimizing entropy relative to a channel or a subalgebra, in: GROUP21, Proc. XXI Int. Coll. on Group Theoretical Methods in Physics, Vol. 1, H. D. Doebner, P. Nattermann, W. Scherer, eds., p. 343, World Scientific, 1997, quant-ph/9701014.Google Scholar
  15. 15.
    S. Hill, W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997), quant-ph/9703041.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Armin Uhlmann
    • 1
  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

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