ar X iv : q ua nt - p h / 02 08 11 1 v 1 1 5 A ug 2 00 2 ENTANGLEMENT , QUANTUM ENTROPY AND MUTUAL INFORMATION

The operational structure of quantum couplings and entanglements is studied and classified for semifinite von Neumann algebras. We show that the classical-quantum correspondences such as quantum encodings can be treated as diagonal semi-classical (d-) couplings, and the entanglements characterized by truly quantum (q-) couplings, can be regarded as truly quantum encodings. The relative entropy of the d-compound and entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the maximum of mutual information over all d-entanglements, and the dimensional entropy, which is achieved at the standard entanglement – true quantum entanglement, coinciding with a d-entanglement only in the case of pure marginal states. The dand qinformation of a quantum noisy channel are respectively defined via the input dand qencodings, and the q-capacity of a quantum noiseless channel is found as the logarithm of the dimensionality of the input algebra. The quantum capacity may double the classical capacity, achieved as the supremum over all d-couplings, or encodings, bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra.