Developments on Quantization and Quantum Information Analysis in terms of Infinite Dimensional Stochastic Analysis

无限维随机分析的量化和量子信息分析进展

基本信息

批准号：
17540136
负责人：
SAITO Kimiaki
金额：
$ 1.15万
依托单位：
Meijo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

We deeply appreciate the grant for scientific research (term : academic years 2005, 2006) from JSPS. In this research, we considered a quantization and a new approach to a quantum information analysis by researching the infinite dimensional stochastic analysis jointly from major fields : probability theory, analysis, non-commutative geometry, number theory and computer science.Main results which we obtained are the following 1) We constructed an infinite dimensional stochastic process associated with the Levy Laplacian on a space based on a stochastic process given by difference between two independent Levy processes. Moreover we gave a necessary and sufficient condition for eigenfunctions of the Levy Laplacian, which has some relation to the quantum decomposition. 2) We can take a nuclear space based on the Levy trace as a domain of the Levy Laplacian and prove that the Levy Laplacian is an infinitesimal generator of an infinite dimensional Wiener process on this space. The state spac … More e of this construction of the stochastic process is new one which is different from the Cesaro Hilbert space introduced by Professor Accardi. 3) Introducing the quantum Levy Laplacian we applied the construction of the infinite dimensional stochastic process generated by the Levy Laplacian in 1) to that of the quantum stochastic process. This implies that the quantum stochastic process can be studied as an infinite dimensional stochastic analysis. The construction of infinite dimensional stochastic processes which the author researched is connected with the quantum information analysis. This result is also connected with the quantum computation. 4) We obtained a relationship between the Levy Laplacian and an infinite dimensional Fractional Ornstein-Uhlenbeck process. This relationship is important to be applied the stochastic analysis based on the Levy Laplacian for the mathematical finance. Moreover we can extend this result to get a relationship between the Laplacian and a general infinite dimensional Ornstein-Uhlenbeck process. We also can obtain some relationship between the quantum Fractional Ornstein-Uhlenbeck process and the quantum Levy Laplacian.By the above research, in particular, we have started a new joint work with Professor Accardi of Volterra Center in University of Rome, and a joint program between Department of Mathematics in our University and the Volterra Center. We also have started a new joint research on the Levy Laplacian on the abstract Wiener space with Professor Kuo of Louisiana State University in USA, which is developed to fruitful results in quantum theory. Less

我们非常感谢JSP的科学研究赠款（学期：学年2005年，2006年）。在这项研究中，我们通过研究来自主要领域的无限尺寸随机分析来考虑一种量化和新方法，从主要领域共同研究：概率理论，分析，非交换性几何学，数字理论和计算机科学。我们获得的米数结果是下面的1）我们通过在差异的过程中构建了一个无限的随机过程，该过程构建了一个差异的过程。此外，我们为Laplacian的本征函数提供了必要和充分的条件，这与量子分解有一定的关系。 2）我们可以根据征费痕迹作为征费拉普拉斯的域名，并证明Laplacian是该空间上无限尺寸维也纳过程的无限发电机。国家空间……这种随机过程的构建更多是新的，它与阿卡迪教授提出的塞萨罗·希尔伯特（Cesaro Hilbert）空间不同。 3）引入量子征费拉普拉斯式，我们应用了征费拉普拉斯在1）中生成的无限尺寸随机过程的构建到量子随机过程的无限尺寸。这意味着可以将量子随机过程作为无限尺寸随机分析进行研究。作者研究的无限尺寸随机过程的构建与量子信息分析有关。该结果还与量子计算相连。 4）我们获得了征费的拉普拉斯和无限尺寸分数Ornstein-Uhlenbeck过程之间的关系。这种关系很重要，必须基于征费拉普拉斯的随机分析作为数学金融。此外，我们可以扩展此结果，以获得Laplacian和一般无限尺寸Ornstein-Uhlenbeck过程之间的关系。我们还可以在量子分数Ornstein-uhlenbeck过程与量子征费拉普拉曲霉之间获得一些关系。特别是，我们已经与罗马大学沃特拉中心的Accardi以及我们大学数学系与Volterra中心的联合计划开始了一项新的联合工作。我们还与美国路易斯安那州立大学的Kuo教授在摘要Wiener Space上开始了一项关于征费拉普拉斯的新联合研究，该研究在量子理论方面富有成果。较少的

项目成果

期刊论文数量（28）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Stochastic process associated with a sum of the Levy Laplacian

与 Levy Laplacian 总和相关的随机过程

DOI：
发表时间：
2005
期刊：
Stochastic Analysis and Applications 23
影响因子：
0
作者：
K.Nishi;K.Saito;A.H.Tsoi;K.Saito;K.Saito
通讯作者：
K.Saito

Infinite Dimensional Harmonic Analysis III

无限维谐波分析 III

DOI：
发表时间：
2005
期刊：
影响因子：
0
作者：
H.Heyer;T.Hirai;T.Kawazoe;K.Saito
通讯作者：
K.Saito

An infinite dimensional Laplacian acting on some class of Levy white noise functionals

作用于某类 Levy 白噪声泛函的无限维拉普拉斯算子

DOI：
发表时间：
2005
期刊：
Proc.Quantum Information 2003 (L. Accardi eds.) (World Scientific) (to appear)
影响因子：
0
作者：
K.Saito;K.Saito;K.Saito;K.Saito
通讯作者：
K.Saito

Quantum Information V

量子信息V