Quantum dynamical entropy revisited

Thomas Hudetz

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 241-251
  • ISSN: 0137-6934

Abstract

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We define a new quantum dynamical entropy for a C*-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.

How to cite

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Hudetz, Thomas. "Quantum dynamical entropy revisited." Banach Center Publications 43.1 (1998): 241-251. <http://eudml.org/doc/208844>.

@article{Hudetz1998,
abstract = {We define a new quantum dynamical entropy for a C*-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.},
author = {Hudetz, Thomas},
journal = {Banach Center Publications},
keywords = {quantum dynamical entropy; -algebra automorphism; invariant state},
language = {eng},
number = {1},
pages = {241-251},
title = {Quantum dynamical entropy revisited},
url = {http://eudml.org/doc/208844},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Hudetz, Thomas
TI - Quantum dynamical entropy revisited
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 241
EP - 251
AB - We define a new quantum dynamical entropy for a C*-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.
LA - eng
KW - quantum dynamical entropy; -algebra automorphism; invariant state
UR - http://eudml.org/doc/208844
ER -

References

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  1. [1] R. Alicki and M. Fannes, Defining Quantum Dynamical Entropy, Lett. Math. Phys. 32 (1994), 75-82. Zbl0814.46055
  2. [2] R. Alicki, J. Andries, M. Fannes and P. Tuyls, An Algebraic Approach to the Kolmogorov-Sinai Entropy, Rev. Math. Phys. 8 (2) (1996), 167-184. Zbl0884.46039
  3. [3] R. Alicki, D. Makowiec, W. Miklaszewski, Quantum Chaos in Terms of Entropy for the Periodically Kicked Top, Phys. Rev. Lett. 77 (1996), 838-841. 
  4. [4] R. Alicki and H. Narnhofer, Comparison of Dynamical Entropies for the Noncommutative Shifts, Lett. Math. Phys. 33 (1995), 241-247. Zbl0836.46059
  5. [5] J. Andries, M. Fannes, P. Tuyls and R. Alicki, The Dynamical Entropy of the Quantum Arnold Cat Map, Lett. Math. Phys. 35 (1995), 375-383. Zbl0842.58054
  6. [6] F. Benatti, T. Hudetz and A. Knauf, Quantum Chaos and Dynamical Entropy, preprint no. 268, SFB 288, TU Berlin, June 1997 (submitted). Zbl0927.37008
  7. [7] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. I / Vol. II (2nd ed.), Springer, New York/Heidelberg/Berlin, 1981/1987. Zbl0463.46052
  8. [8] A. Connes, H. Narnhofer and W. Thirring, Dynamical Entropy of C* Algebras and von Neumann Algebras, Commun. Math. Phys. 112 (1987), 691-719. Zbl0637.46073
  9. [9] A. Connes and E. Størmer, Entropy for Automorphisms of I I 1 von Neumann Algebras, Acta Math. 134 (1975), 289-306. Zbl0326.46032
  10. [10] V. Ya. Golodets and E. Størmer: Entropy of C*-Dynamical Systems Defined by Bitstreams, preprint no. 18, Dept. of Mathematics, Univ. of Oslo, August 1996. 
  11. [11] T. Hudetz, A `Hybrid' State-Dependent Dynamical Entropy for C*-Algebra Automorphisms, preprint, Univ. of Vienna (to be submitted). Zbl0927.46049
  12. [12] A. N. Kolmogorov, A New Metric Invariant of Transitive Systems and Automorphisms of Lebesgue Spaces, Dokl. Akad. Nauk SSSR 119 (1958), 861-864. Ya. G. Sinai, On the Concept of Entropy of a Dynamical System, Dokl. Akad. Nauk SSSR 124 (1959), 768-771. Zbl0083.10602
  13. [13] H. Narnhofer, E. Størmer and W. Thirring, C* Dynamical Systems for Which the Tensor Product Formula for Entropy Fails, Ergod. Th. & Dynam. Sys. 15 (1995), 961-968. Zbl0832.46059
  14. [14] H. Narnhofer and W. Thirring, C*-Dynamical Systems that Asymptotically are Highly Anticommutative, Lett. Math. Phys. 35 (1995), 145-154. Zbl0833.46051
  15. [15] G. Lindblad, Dynamical Entropy for Quantum Systems, in: Quantum Probability and Applications Vol. III, L. Accardi and W. von Waldenfels (eds.), Springer LNM 1303, Berlin, 1988, pp. 183-191. 
  16. [16] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer Texts and Monographs in Physics, Berlin/Heidelberg/New York, 1993. Zbl0891.94008
  17. [17] R. T. Powers and G. L. Price, Binary Shifts on the Hyperfinite I I 1 Factor, Contemporary Math. 145 (1993), 453-464. Zbl0813.46050
  18. [18] R. T. Powers and G. L. Price, Cocycle Conjugacy Classes of Shifts on the Hyperfinite I I 1 Factor, J. Funct. Anal. 121 (1994), 275-295. Zbl0824.46068
  19. [19] G. L. Price, The Entropy of Rational Powers Shifts, preprint, Department of Mathematics, United States Naval Academy (Annapolis, Maryland), 1996. 
  20. [20] E. Størmer and D. Voiculescu, Entropy of Bogoliubov Automorphisms of the Canonical Anticommutation Relations, Commun. Math. Phys. 133 (1990), 521-542. Zbl0743.46051
  21. [21] P. Tuyls, Towards Quantum Kolmogorov-Sinai Entropy, Ph.D. Thesis, Univ. Leuven, 1997. Quantization: Non-commutative Dynamical Entropy, Rep. Math. Phys. 38 (1996), 437-442. Zbl0897.60100
  22. [22] D. V. Voiculescu, Dynamical Approximation Entropies and Topological Entropy in Operator Algebras, Commun. Math. Phys. 170 (1995), 249-281. Zbl0824.46079

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