Information Geometry for Quantum Systems and Its Applications

点击翻译按钮获取中文摘要

This research project aims at deepening information geometry for quantum systems from a statistical/information-theoretical viewpoint and also at investigating new aspects of statistical/information-theoretical problems for quantum systems from a geometrical viewpoint, mainly focusing upon quantum estimation theory and quantum relative entropy. Main results are as follows.1.We have provided various quantum analogues of the Fisher metric and the (alpha=1,-1)-connections on the space of faithful quantum states with a unifying theoretical framework and have studied the general theory of quantum information-geometrical structure as well as physical/information-theoretical significance of some specific structures.2.We have demonstrated several information-geometrical aspects of a manifold of quantum pure states, among which are a close relation between the dually flat structure and the Kaehler structure for a quantum exponential family of pure states and an extension of the RLD metric on the complexified cotangent space.3.The Boltzmann machine is a kind of stochastic neural network and is known to be fit to the framework of information geometry by the fact that its equilibrium distributions form an exponential family. We have studied a quantum extension of Boltzmann machine from an information-geometrical viewpoint to clarify analogy and difference between the classical and quantum cases.4.We have also investigated several related subjects on estimation theory for quantum channels, classical and quantum information theory and information geometry for stochastic processes.