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Infinite Dimensional Analysis and Quantum Probability Theory

无限维分析和量子概率论

基本信息

批准号：
09640178
负责人：
ACCARDI Luigi
金额：
$ 1.92万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

(1) Stochastic Limit of Quantum Theory Through the stochastic limit quantum noises and quantum stochastic differential equations are derived from canonical Hamiltonian models. This *ethod is applied to a model involving strong nonlinear interaction to obtain a new type of quantum noise. For example, a standard model of QED without dipole approximation yields a singular type of noncrossing diagrams and nonlinear deformation of Wigner's semicircle law. The concept of an interacting Fock module is introduced in order to unify this kind of phenomena.(2) Nonlinear Extension of Classical and Quantum Stochastic Calculus Motivated by (1) , we studied higher powers of quantum white noises which are considered as the first class of an infinite hierarchy of noises. Tha associated Ito formula is proved with renormalization. Moreover, unique existence of solutions is proved for a class of normal-ordered white noise equations involving higher powers of quantum white noises and a relation to quantum stochastic differential equations is established.(3) Central Limit Theorems Within the framework of algebraic probability theory, the existing concepts of independence have been unified and the new notion of singleton independence is introduced. The associated central limit theorem is proved and the limit stochastic process is obtained. This result is related to orthogonal polynomials, their q-deformations, and Gaussianization * probability measures.(4) Quantum Markov Chains Classification of quantum Markov states and lifting problems in quantum communication channels are discussed. An interesting connection between basic algorithyms appearing in quantum computing and quantum Markov chains is investigated.(5) Others Some conrete nonlinear models are discussed in detail in connection with quantum entropy and quantum communication. Questions in foundation of quantum theory related to Bell inequality and EPR paradox are clarified from the standpoint of algebraic probability theory.

(1)量子理论的随机极限从正则哈密顿模型出发，通过随机极限导出了量子噪声和量子随机微分方程。将此方法应用于强非线性相互作用模型，得到了一种新的量子噪声。例如，没有偶极近似的QED标准模型产生了奇异型的不相交图和Wigner周期律的非线性变形。为了统一这类现象，引入了相互作用Fock模的概念。(2)经典和量子随机微积分的非线性推广受（1）的启发，我们研究了量子白色噪声的高次幂，它被认为是无限噪声族的第一类。用重整化方法证明了相应的Ito公式。进一步证明了一类含有高次幂量子白色噪声的正序白色噪声方程解的存在唯一性，并建立了与量子随机微分方程的联系. (3)中心极限定理在代数概率论的框架内，统一了已有的独立性概念，引入了新的单点独立性概念。证明了相应的中心极限定理，得到了极限随机过程。这个结果与正交多项式、它们的q变形和高斯化 * 概率测度有关。(4)量子马尔可夫链讨论了量子马尔可夫态的分类和量子通信信道中的提升问题。研究了量子计算中出现的基本算法和量子马尔可夫链之间的一种有趣的联系。(5)其他结合量子熵和量子通信，详细讨论了几种具体的非线性模型。从代数概率论的角度阐明了量子理论基础中与贝尔不等式和EPR佯谬有关的问题。