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

我们深入了解JSP的科学研究（学期：2005年，2005年），通过研究无限的维度随机分析，共同研究了量子信息分析的新方法。数字理论和计算机有限的代数随机措施与laplacian相关的空间基于独立的征量解压缩。拉普拉斯（Laplacian）是状态空格上无限型原始质量的无限发电机...他是新的过程与accardi教授引入的cesaro hilbert空间不同。征收laplacian在1个uantum的随机过程中，这意味着tothastic sisis可以在无限的尺寸随机过程中。无限的Ornstein-uhleck ESS。 Laplacian。通过上述研究，尤其是罗马大学的Olterra中心，HABE还与美国路易斯安那州立大学的Kuo一起开始了一项关于摘要Wiener空间的联合研究。

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