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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

关于各种主题的统计调查如下。(1)从统计模型与统计量之间的关系出发，讨论了模型的性质，得到了关于各种统计量行为的一些有趣的结果。(2)在金融统计理论、时间序列及其应用中，统计过程被证明是渐近有用的。(3)在实验设计及其相关领域，数学结构是明确的组合程序，并打算应用于实际问题的结果得到。(4)在统计序贯推理中，提出了几种序贯过程，并详细讨论了它们的性质。给出了渐近效率。(5)尝试建立生物统计学的数学基础理论，并指出统计推断程序在其中的重要作用。特别是生物测定法、评分法等，在临床上是有效的。 ...更多信息 L. (6)阐明了量子力学中的非定域性与统计推断之间的关系，证明了统计推断在量子估计和量子检验中的有效性。此外，它被认为是作为具体物理现象的理论基础发挥重要作用。(7)在统计推断中，从与传统Bahadur有效性不同的角度研究了相合估计的尾概率下界的一阶和二阶渐近有效性.为了统一非参数检验和参数检验，建立了相应的数学模型，提出了基于谱密度矩阵估计量的检验统计量，并给出了它们的渐近性质。(8)在一族非参量量子态下，研究了量子态的状态估计、量子态的预测、量子信息几何和量子态的判别问题，得到了一些新的有趣结果.为此，双方举行了多次研讨会，并进行了积极的讨论和信息交流。他们的研究结果汇总成册。少