Note on Transmitted complexity for Quantum Compound Systems

Ohya proposed the information dynamics synthesizing several ways of studying complex systems. In information dynamics, there are two types of complexities, one is a complexity of state representing system itself and another is a transmitted complexity between two systems. Entropies in classical and quantum systems are examples of these complexities of information dynamics. Transmitted complexity is an important tool to analyse the efficiency of information transmission in communication processes. In order to treat a flow of dynamical process, dynamical entropies were introduced in not only classical but also quantum systems. \qquad Base on the transition expectation introduced by Accardi to study quantum Markov process, the KOW entropy for completely positive (CP) maps was defined in [3]. The generalized AOW and the AF entropis was constructed by the KOW entropy. The compound states are important tool to define the transmitted entropy [19,22]. The transmitted complexity associated with the separable compound states is defined by using the generalized AOW entropy in [14,23]. \qquad In the talk, we briefly review the generalized AOW entropy formulated by KOW entropy and we define a transmitted complexity (dynamical mutual entropy) given by a modified compound states and we prove the fundamental inequalities of the transmitted complexity by means of the independent dynamical systems.