  The purpose of this paper is to survey some topics on mathematical
foundations of quantum information developed mainly by the present author and
co-workers for the last three decades. The topics include an axiomatic
construction of quantum measurement theory based on completely positive
map-valued measures, a universally valid new formulation of the uncertainty
principle for error and disturbance in quantum measurements, the
Wigner-Araki-Yanase limit of quantum measurements, the accuracy limit of
quantum computing based on conservation laws, and a quantum interpretation
based on quantum set theory.
