In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch 13 and Connes–Stormer 11. After that, several approaches for introducing the quantum dynamical entropy were proposed 10,[Formula: see text3,Formula: see text8,Formula: see text39,Formula: see text15,Formula: see text44,Formula: see text9,Formula: see text27,Formula: see text28,Formula: see text2,Formula: see text19,Formula: see text45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy 22 and the Ohya mutual entropy 24. These entropies were extended to S-mixing entropy by Ohya 26,[Formula: see text27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.
Noboru Watanabe (Sun,) studied this question.