Information Geometry for Quantum Systems and Its Applications

本项目旨在从统计/信息理论的角度深化量子系统的信息几何，从几何的角度研究量子系统的统计/信息理论问题的新方面，主要集中在量子估计理论和量子相对熵。主要研究结果如下：我们在一个统一的理论框架下提供了Fisher度规和忠实量子态空间上的(α =1,-1)-连接的各种量子类似物，并研究了量子信息的一般理论——几何结构以及某些特定结构的物理/信息理论意义。我们证明了量子纯态流形的几个信息几何方面，其中包括纯态量子指数族的对偶平面结构和Kaehler结构之间的密切关系，以及RLD度量在复切空间上的扩展。玻尔兹曼机是一种随机神经网络，它的平衡分布呈指数族，因而适合于信息几何的框架。从信息几何的角度研究了玻尔兹曼机的量子扩展，阐明了经典情况与量子情况的异同。我们还研究了量子信道估计理论、经典和量子信息理论以及随机过程的信息几何等相关课题。

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.