Fundamentals of Quantum Mutual Entropy and Capacity

The study of mutual entropy (information) and capacity in classical system was extensively done after Shannon by several authors like Kolmogorov [12] and Gelfand [7]. In quantum systems, there have been several definitions of the mutual entropy for classical input and quantum output [5, 8, 9, 14]. In 1983, the author defined [21] the fully quantum mechanical mutual entropy by means of the relative entropy of Umegaki [35], and he extended it [23] to general quantum systems by the relative entropy of Araki [3] and Uhlmann [34]. When the author introduced the quantum mutual entropy, he did not indicate that it contains other definitions of the mutual entropy including classical one, so that there exist several misunderstandings for the use of the mutual entropy (information) to compute the capacity of quantum channels. Therefore in this note we point out that our quantum mutual entropy generalizes others and where the misuse occurs.