  In quantum information processing it may be possible to have efficient
computation and secure communication beyond the limitations of classical
systems. In a fundamental point of view, however, evolution of quantum systems
by the laws of quantum mechanics is more restrictive than classical systems,
identified to a specific form of dynamics, that is, unitary transformations
and, consequently, positive and completely positive maps to subsystems. This
also characterizes classes of disallowed transformations on quantum systems,
among which positive but not completely maps are of particular interest as they
characterize entangled states, a general resource in quantum information
processing. Structural physical approximation offers a systematic way of
approximating those non-physical maps, positive but not completely positive
maps, with quantum channels. Since it has been proposed as a method of
detecting entangled states, it has stimulated fundamental problems on
classifications of positive maps and the structure of Hermitian operators and
quantum states, as well as on quantum measurement such as quantum design in
quantum information theory. It has developed efficient and feasible methods of
directly detecting entangled states in practice, for which proof-of-principle
experimental demonstrations have also been performed with photonic qubit
states. Here, we present a comprehensive review on quantum information
processing with structural physical approximations and the related progress.
The review mainly focuses on properties of structural physical approximations
and their applications toward practical information applications.
