Entanglement, quantum entropy and mutual information

The operational structure of quantum couplings and entanglements is studied and classified for semi–finite 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 d– and q–information of a quantum noisy channel are, respectively, defined via the input d– and q–encodings, and the q–capacity of a quantum noiseless channel is found to be 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.

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