  The study of mutual entropy (information) and capacity in classica l system
was extensively done after Shannon by several authors like Kolmogor ov and
Gelfand. In quantum systems, there have been several definitions of t he mutual
entropy for classical input and quantum output. In 1983, the autho r defined
the fully quantum mechanical mutual entropy by means of the relati ve entropy
of Umegaki, and it has been used to compute the capacity of quant um channel
for quantum communication process; quantum input-quantum output.
  Recently, a correlated state in quantum syatems, so-called quantum entangled
state or quantum entanglement, are used to study quntum information, in part
icular, quantum computation, quantum teleportation, quantum cryptography.
  In this paper, we mainly discuss three things below: (1) We point out the di
fference between the capacity of quantum channel and that of classical-quant
um-classical channel. (2) So far the entangled state is merely defined as a
non-separable state, we give a wider definition of the entangled state and c
lassify the entangled states into three categories. (3) The quantum mutual e
ntropy for an entangled state is discussed. The above (2) and (3) are a join t
work with Belavkin.
