Note on quantum dynamical entropies

Classical dynamical entropy is an important tool to analyse the efficiency of information transmission in communication processes. Quantum dynamical entropy was first studied by Connes, Størmer and Emch. Since then, there have been many attempts to formulate or compute the dynamical entropy for some models. Here we review four formulations due to 1.(a) Connes, Narnhofer and Thirring, 2.(b) Ohya, 3.(c) Accardi, Ohya and Watanabe, 4.(d) Alicki and Fannes. We consider mutual relations between these formulations and we show some concrete computations for a model