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

我们非常感谢JSPS的科学研究资助（期限：2005，2006学年）。在这项研究中，我们考虑了量子化和量子信息分析的新方法，从主要领域联合研究无限维随机分析：概率论，分析，非对易几何，数论和计算机科学。我们得到的主要结果如下：1）基于由两个独立Levy过程之差给出的随机过程，构造了一个与Levy Laplacian相关的无穷维随机过程。此外，我们给出了Levy Laplacian的本征函数的一个充要条件，它与量子分解有一定的关系。2)我们可以把一个基于Levy迹的核空间作为Levy Laplacian的定义域，并证明Levy Laplacian是这个空间上无限维Wiener过程的无穷小生成元。状态空间 ...更多信息 e是一种新的随机过程构造，它不同于Accardi教授提出的塞萨罗Hilbert空间。3)通过引入量子Levy Laplacian，我们将1）中由Levy Laplacian生成的无穷维随机过程的构造应用于量子随机过程的构造。这意味着量子随机过程可以作为无限维随机分析来研究。作者所研究的无穷维随机过程的构造与量子信息分析密切相关。这个结果也与量子计算有关。4)得到了Levy Laplacian与无穷维分数阶Ornstein-Uhlenbeck过程之间的关系。这一关系对于基于Levy Laplacian的随机分析在数理金融中的应用具有重要意义。此外，我们可以推广这个结果，得到拉普拉斯算子和一般的无穷维Ornstein-Uhlenbeck过程之间的关系。通过以上研究，我们与罗马大学沃尔泰拉中心的Accardi教授开始了一项新的合作工作，并开始了我校数学系与沃尔泰拉中心的合作项目。我们还与美国路易斯安那州立大学的郭教授开始了关于抽象Wiener空间上的Levy Laplacian的新的联合研究，在量子理论方面取得了丰硕的成果。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