Some aspects of complexities for quantum systems

We are interested to study the dynamics of state change or the complexity of state for several systems. Information dynamics (ID) is a new concept introduced by Ohya to construct a theory under the ID´s framework by synthesizing these investigating schemes. In ID, a complexity of state describing system itself and a transmitted complexity between two systems are used. One of the example of these complexities are the Shannon´s entropy and the mutual entropy (information) in classical entropy theory. In quantum entropy theory, it was known that the von Neumann entropy and the Ohya mutual entropy relate to these complexities. Recently, several mutual entropy type measures (the Lindblad-Nielsen entropy and the Coherent entropy) were proposed by means of the entropy exchange for an input state and a channel. The quantum capacity is defined by taking the supremum for the Ohya mutual entropy. The main purpose of this paper is to compare with these mutual entropy type complexities in order to find most congruous one for studying the quantum communication process. In this paper, we calculate the quantum capacity for some simple models for discussing the information transmission for OOK and PSK modulations.