  The mathematical structure of quantum entanglement is studied and classified
from the point of view of quantum compound states. We show that t he
classical-quantum correspondences such as encodings can be treated as dia gonal
(d-) entanglements. The mutual entropy of the d-compound and entangled states
lead to two different types of entropies for a given quantum state: t he von
Neumann entropy, which is achieved as the supremum of the information over all
d-entanglements, and the dimensional entropy, which is achieved at the standard
entanglement, the true quantum entanglement, coinciding with a d-entanglement
only in the case of pure marginal states. The q-capacity of a quantum noiseless
channel, defined as the supremum over all entanglements, i s given by the
logarithm of the dimensionality of the input algebra. It doub les the classical
capacity, achieved as the supremum over all d-entanglement s (encodings), which
is bounded by the logarithm of the dimensionality of a maximal Abelian
subalgebra.
