A mathematical foundation of quantum information and quantum computer -on quantum mutual entropy and entanglement-

TL;DR

This paper explores the mathematical foundations of quantum information, focusing on quantum mutual entropy, entanglement, and their implications for quantum channel capacity, offering new classifications and broader definitions of entangled states.

Contribution

It introduces a wider definition and classification of entangled states and discusses quantum mutual entropy for these states, expanding theoretical understanding.

Findings

Distinction between quantum and classical-quantum channel capacities

New classification of entangled states into three categories

Discussion of quantum mutual entropy for entangled states

Abstract

The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entangled state or quantum entanglement, are used to study quntum information, in part icular, quantum computation, quantum teleportation, quantum cryptography. In this paper, we mainly discuss three things below: (1) We point out the di fference between the capacity of quantum channel and that of classical-quant um-classical channel. (2) So far the entangled state is merely defined as a non-separable state, we give a wider definition of the entangled state and c lassify the entangled states into three categories. (3) The quantum mutual e ntropy for an entangled state is discussed. The above (2) and (3) are a join t work with Belavkin.