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Information Geometry for Quantum Systems and Its Applications

量子系统信息几何及其应用

基本信息

批准号：
15500004
负责人：
NAGAOKA Hiroshi
金额：
$ 2.3万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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.

本研究项目旨在从统计/信息论的角度深化量子系统的信息几何，并从几何的角度研究量子系统统计/信息论问题的新方面，主要集中在量子估计理论和量子相对熵。主要结果如下：1.我们以统一的理论框架提供了Fisher度量和忠实量子态空间上的(alpha=1,-1)-连接的各种量子类比，并研究了量子信息几何结构的一般理论以及一些特定结构的物理/信息理论意义。2.我们证明了多种量子纯态的几个信息几何方面，其中它们是纯态量子指数族的对偶平面结构和凯勒结构之间的密切关系，也是复余切空间上 RLD 度量的扩展。 3. 玻尔兹曼机是一种随机神经网络，由于其平衡分布形成指数族，因此适合信息几何框架。我们从信息几何的角度研究了玻尔兹曼机的量子扩展，以阐明经典情况和量子情况之间的类比和区别。4.我们还研究了量子通道估计理论、经典和量子信息论以及随机过程信息几何等几个相关课题。