The Foundation of Mathematical Statistics on Quantum Inference and Its Applications

量子推理的数理统计基础及其应用

基本信息

批准号：
14204006
负责人：
AKAHIRA Masafumi
金额：
$ 29.7万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

The statistical investigation on various themes was done as follows. (1) Involving a relationship between statistical models and statistics, the properties of the models were discussed and some interesting results on the behavior of various statistics are obtained. (2) In the theory of statistics to finance, time series and their applications, statistical procedures were shown to be asymptotically useful. (3) In the experimental design and its related area, the mathematical structure is clarified by combinatorial procedures, and results intended to apply to practical problems were obtained. (4) In statistical sequential inference, some sequential procedures were proposed, and their properties were discussed in details. The asymptotic efficiencies were shown. (5) The construction of mathematically fundamental theory of biostatistics is tried, and statistically inferential procedures are shown to play an important role. In particular, bioassay test, score test etc. were shown to be usefu … More l. (6) The relationship between non-locality in quantum mechanics and statistical inference is clarified, and inferential procedures is also shown to be efficient in quantum estimation and quantum test. Further, it is recognized to play an important role as the theoretical base of concrete physical phenomena. (7) In statistical inference, on the lower bound for tail probabilites of consistent estimators, the first and second order asymptotic efficiencies are investigated from a different viewpoint from conventional Bahadur efficiency. And in order to unify both of non-parametric and parametric tests, the mathematical setup was done, new test statistics based on estimators of spectral density matrics were proposed, and their asymptotic properties are derived. (8) Under a family of non-parametric quantum states, state estimation, prediction of quantum state, quantum information geometry and discrimination problem on quantum states are trated, and new interesting results were obtained. Many symposium on the above were held and active discussion and mutual exchange of information were also done. Their results were summarized as a volume. Less

对各个主题的统计调查如下：（1）涉及统计模型和统计之间的关系，讨论了模型的属性，并获得了有关各种统计行为的一些有趣的结果。 (3) 在实验设计及其相关领域，通过组合程序阐明了数学结构，并获得了适用于实际问题的结果 (4) (5)尝试了生物统计学数学基础理论的构建，并证明了特别的推理程序的重要作用。特别是，生物测定测试、测试分数等被证明是有用的……更多 l. (6) 阐明了量子力学中的非局域性和统计推理之间的关系，并且推理程序也被证明是有效的。此外，它被认为作为具体物理现象的理论基础发挥着重要作用。 (7) 在统计推断中，关于一致估计量的尾部概率的下界，一阶和二阶渐近效率。从与传统巴哈杜尔效率不同的角度进行研究，并且为了统一非参数和参数测试，完成了数学设置，提出了基于谱密度矩阵估计的新测试统计量。 (8)对非参数量子态族下的状态估计、量子态预测、量子信息几何和量子态判别问题进行了研究，取得了许多有趣的新成果。并进行了积极的讨论和相互交流，成果总结为一卷。