ArticleOriginal scientific text
Title
Quantum dynamical entropy revisited
Authors 1
Affiliations
- Institute of Mathematics, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria
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.
Bibliography
- R. Alicki and M. Fannes, Defining Quantum Dynamical Entropy, Lett. Math. Phys. 32 (1994), 75-82.
- 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.
- R. Alicki, D. Makowiec, W. Miklaszewski, Quantum Chaos in Terms of Entropy for the Periodically Kicked Top, Phys. Rev. Lett. 77 (1996), 838-841.
- R. Alicki and H. Narnhofer, Comparison of Dynamical Entropies for the Noncommutative Shifts, Lett. Math. Phys. 33 (1995), 241-247.
- 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.
- F. Benatti, T. Hudetz and A. Knauf, Quantum Chaos and Dynamical Entropy, preprint no. 268, SFB 288, TU Berlin, June 1997 (submitted).
- 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.
- A. Connes, H. Narnhofer and W. Thirring, Dynamical Entropy of C* Algebras and von Neumann Algebras, Commun. Math. Phys. 112 (1987), 691-719.
- A. Connes and E. Størmer, Entropy for Automorphisms of
von Neumann Algebras, Acta Math. 134 (1975), 289-306. - 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.
- T. Hudetz, A `Hybrid' State-Dependent Dynamical Entropy for C*-Algebra Automorphisms, preprint, Univ. of Vienna (to be submitted).
- 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.
- 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.
- H. Narnhofer and W. Thirring, C*-Dynamical Systems that Asymptotically are Highly Anticommutative, Lett. Math. Phys. 35 (1995), 145-154.
- 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.
- M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer Texts and Monographs in Physics, Berlin/Heidelberg/New York, 1993.
- R. T. Powers and G. L. Price, Binary Shifts on the Hyperfinite
Factor, Contemporary Math. 145 (1993), 453-464. - R. T. Powers and G. L. Price, Cocycle Conjugacy Classes of Shifts on the Hyperfinite
Factor, J. Funct. Anal. 121 (1994), 275-295. - G. L. Price, The Entropy of Rational Powers Shifts, preprint, Department of Mathematics, United States Naval Academy (Annapolis, Maryland), 1996.
- E. Størmer and D. Voiculescu, Entropy of Bogoliubov Automorphisms of the Canonical Anticommutation Relations, Commun. Math. Phys. 133 (1990), 521-542.
- 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.
- D. V. Voiculescu, Dynamical Approximation Entropies and Topological Entropy in Operator Algebras, Commun. Math. Phys. 170 (1995), 249-281.