Mathematical foundations of quantum information: Measurement and foundations

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.