Note on entropy type complexity for transmission Process

The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself. It was extended by Ohya for $C^*$-systems before CNT entropy. The quantum relative entropy was first defined by Umegaki for $\sigma$-finite von Neumann algebras, which was extended by Araki and Uhlmann for general von Neumann algebras and $*$-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to formulate the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this talk, we briefly review the entropic complexities for classical and quantum systems. We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently.