A MATHEMATICAL FOUNDATION OF QUANTUM INFORMATION AND QUANTUM COMPUTER -on quantum mutual entropy and entanglement-

The study of mutual entropy (information) and capacity in classical system was extensively done after Shannon by several authors like Kolmogorov [16] and Gelfand [10]. In quantum systems, there have been several definitions of the mutual entropy for classical input and quantum output [5, 11, 12, 17]. In 1983, the author defined [21] the fully quantum mechanical mutual entropy by means of the relative entropy of Umegaki[32], and it has been used to compute the capacity of quantum channel for quantum communication process; quantum inputquantum output [25]. Recently, a correlated state in quantum syatems, so-called quantum entangled state or quantum entanglement, are used to study quntum information, in particular, quantum computation, quantum teleportation, quantum cryptography [6, 7, 8, 9, 14, 15, 29, 30]. In this paper, we mainly discuss three things below:(1) We point out the difference between the capacity of quantum channel and that of classicalquantum-classical channel followed from [28]. (2) So far the entangled state is merely defined as a non-separable state, we give a wider definition of the entangled state and classify the entangled states into three categories. (3)The quantum mutual entropy for an entangled state is discussed. The above (2) and (3) are a joint work with Belavkin[6] .