Abstract
We compute the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of shift automorphism of generalized quantum Markov chains as defined by Accardi and Frigerio. For any generalized quantum Markov chain defined via a finite set of conditional density amplitudes, we show that the dynamical entropy is equal to the mean entropy.
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References
Accardi, L. and Frigerio, A., Markov cocycles,Proc. R. Ir. Acad. 83A, 251–269 (1983).
Accardi, L. and Watson, G. S., Quantum random walks, inLecture Notes in Math. 1936, Springer-Verlag, New York, 1989, pp. 73–78.
Besson, O., On the entropy of quantum Markov states, inLecture Notes in Math. 1136, Springer-Verlag, New York, 1985, pp. 81–89.
Bratteli, O. and Robinson, D. W.,Operator Algebras and Quantum Statistical Mechanics. I, II, Springer-Verlag, New York, 1981.
Connes, A., Narnhofer, H., and Thirring, W., Dynamical entropy ofC *-algebras and von Neumann algebras,Comm. Math. Phys. 112, 691–719 (1987).
Connes, A. and Størmer, E., Entropy of automorphisms ofII 1 von Neumann algebras,Acta Math. 134, 289–306 (1975).
Kolmogorov, A. N.,Dokl. Akad. Nauk. 119, 861 (1958).
Narnhofer, H. and Thirring, W., Dynamical entropy of quasi-free automorphisms,Lett. Math. Phys. 14, 89–96 (1987).
Narnhofer, H. and Thirring, W., Dynamical entropy of quantum systems and their Abelian counterpart, to appear.
Ohya, M. and Petz, D.,Quantum Entropy and Its Use, Springer-Verlag, New York, 1993.
Park, Y. M. and Shin, H. H., Dynamical entropy of quasi-local algebras in quantum statistical mechanics,Comm. Math. Phys. 144, 149–161 (1992).
Park, Y. M. and Shin, H. H., Dynamical entropy of space translations of CAR and CCR algebras with respect to quasi-free states,Comm. Math. Phys. 152, 497–537 (1993).
Quasthoff, U., Shift automorphisms of the hyperfinite factor,Math. Nachtrichten 131, 101–106 (1987).
Sinai, Ya. G.,Dokl. Akad. Nauk., 768 (1959).
Størmer, E., Entropy of some automorphisms of theII 1-factor of the free group in infinite number of generators,Invent. Math. 110, 63–73 (1992).
Størmer, E., Entropy in operator algebra, Preprint (1992).
Størmer, E. and Voiculescu, D., Entropy of Bogoliubov automorphisms of the canonical anticommutation relations,Comm. Math. Phys. 133, 521–542 (1990).
Walters, P.,An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer-Verlag, New York, 1982.
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Research supported in part by the Basic Science Research Program, Korean Ministry of Education, 1993–1994.