Comparing quantum dynamical entropies

Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as ALF (Alicki, Lindblad, Fannes). Therefore, a natural question arises whether there is a relation between all these different notions. In this paper we will indicate that the CNT entropy turns out to be smaller than the ALF dynamical entropy