Quantum Entanglements and Entangled Mutual Entropy

The mathematical structure of quantum entanglement is studied and classified from the point of view of quantum compound states. We show that the classical-quantum correspondences such as encodings can be treated as diagonal (d-) entanglements. The mutual entropy of the d-compound and entangled states lead to two different types of entropies for a given quantum state: the 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, is given by the logarithm of the dimensionality of the input algebra. It doubles the classical capacity, achieved as the supremum over all d-entanglements (encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra.