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

我们非常感谢JSPS的科研资助（任期：2005、2006学年）。本研究结合概率论、分析学、非交换几何、数论和计算机科学等主要领域对无限维随机分析进行了研究，提出了量子信息分析的量子化和新方法。主要研究结果如下：(1)基于由两个独立Levy过程之差给出的随机过程，在空间上构造了一个与Levy拉普拉斯算子相关的无限维随机过程。此外，我们还给出了与量子分解有关的Levy拉普拉斯本征函数的一个充分必要条件。2)我们可以取一个基于Levy迹的核空间作为Levy Laplacian的定义域，并证明Levy Laplacian是该空间上一个无限维Wiener过程的无穷小发生器。3)引入量子Levy Laplacian，我们将1)中由Levy Laplacian生成的无限维随机过程的构造应用到量子随机过程的构造中。这意味着量子随机过程可以作为一个无限维的随机分析来研究。作者所研究的无限维随机过程的构造与量子信息分析有关。这一结果也与量子计算有关。4)得到了Levy Laplacian与无限维分数阶Ornstein-Uhlenbeck过程的关系。这一关系对于应用基于Levy Laplacian的数学金融随机分析具有重要意义。此外，我们可以推广这一结果，得到拉普拉斯过程与一般无限维Ornstein-Uhlenbeck过程之间的关系。我们还得到了量子分数阶Ornstein-Uhlenbeck过程与量子Levy Laplacian之间的关系。特别是通过上述研究，我们与罗马大学Volterra中心的Accardi教授开展了一项新的联合工作，以及我校数学系与Volterra中心的联合项目。我们还与美国路易斯安那州立大学的郭教授共同开展了抽象Wiener空间上的Levy Laplacian的新研究，并在量子理论方面取得了丰硕成果。少

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