Note on Entropy for Quantum Open Systems

In order to treat several complicated systems, it is important to study the dynamics of state change and the complexity of states of systems. Information Dynamics introduced by Ohya is a new concept synthesizing the research schemes of several complicated systems. In ID, there are two types of complexities, (1) a complexity of state describing system itself and (2) a transmitted complexity between two systems. Classical and quantum entropies are the example of the complexities of those complexites. Based on the Umegaki's relative entropy of Umegaki and the compound state, the quantum mutual entropy was defined by Ohya in 1983, and it was extended to general quantum systems by using the relative entropy of Araki and Uhlmann. One can discuss the coding theorems by means of the mean entropy and the mean mutual entropy defined by the dynamical entropy. In this talk, we will discuss about entropy of open system dynamics.