Quantum proper entropies and additive capacities of quantum information channels

An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.