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

我们对JSPS提供的科研经费(任期：2005、2006学年)深表赞赏。本文从概率论、分析、非对易几何、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，给出了量子信息分析的一种量子化方法和一种新的分析方法。主要结果如下：1)基于两个独立的Levy过程之间的差给出的一个随机过程，我们在空间上构造了一个与Levy Laplace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李维-拉普拉斯本征函数的一个充要条件。2)我们可以把一个基于Levy迹的核空间看作是Levy Laplacian的一个区域，并且证明了Levy Laplace是该空间上无限维Wiener过程的无穷小生成元。国家空间委员会…这种随机过程的构造更多的是一种不同于Accardi教授引入的Cesaro Hilbert空间的新构造。3)引入量子Levy Laplace算子，将由Levy Laplace算子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无限维随机分析来研究。作者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过程之间的关系。这一关系对于基于拉普拉斯函数的随机分析在数学金融中的应用具有重要意义。此外，我们还可以推广这一结果，得到拉普拉斯过程与一般无限维Ornstein-Uhlenbeck过程之间的关系。特别是通过上述研究，我们与罗马大学Volterra中心的Accardi教授开始了新的合作，并与我校数学系与Volterra中心合作开展了一个新的项目。我们还与美国路易斯安那州立大学的郭教授开始了一项关于抽象Wiener空间上的Levy Laplacian的新的联合研究，这项研究在量子理论方面取得了丰硕的成果。较少

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