深圳大学四十周年校庆暨数学学科四十周年庆

荔园学者Colloquium第二十七期

讲座题目:Limit Theorems for Three-State Quantum Walks

主讲人：陈增敬 教授（山东大学）

讲座时间：2023年6月18日9：00-10：00

讲座地点：深圳大学粤海校区国际会议厅大厅

内容概述：Normal distribution occupy a “central” position in probability theory. The classical central limit theorem indicates that a group of random variables will uniformly converge to the normal distribution under certain conditions. An important property of normal distribution is that its density function has only one peak. A natural question is whether normal distributions still occupy a “central” position in quantum theory. To answer this question, in this paper, we investigate several central limit theorems for a discrete-time three-state quantum walks on the real line. We introduce a set of special coin update operators—the rotation matrix, whose degrees of freedom include rotation axis and angle. We obtain their explicit forms of the limit distribution for specific axes and angles and we also derive their simulating distribution curves through computer for unsolvable general unitary matrices. The results reveal that three-state quantum walks exhibit three interesting phenomena that differ from two-state quantum walks. First, no unified form of the limit distribution exists for three-state quantum walks. Second, the probability density of two-state quantum walks has at most two peaks, whereas the probability density of three-state quantum walks could have three peaks or even four. Third, we prove that under some very special initial states, the limit distributions of two-state quantum walks could degenerate into the classical normal or uniform distributions, thus, the normal distribution is just a very special form of limit distributions in quantum walks. In this sense, normal distribution has lost its “central” position in quantum framework.

主讲人简介：陈增敬教授，现任中泰证劵金融研究院院长、教育部教学指导委员会统计学副主任委员、国家杰出青年科学基金获得者。曾任山东大学数学学院院长。曾获国家自然科学奖二等奖(独立，2015年)和孙冶方经济科学奖(首位，2011年)。先后在国内外重要学术刊物上发表论文多篇，对不完备市场中的资产定价问题的研究成果被国际著名学者多次引用和推广。特别是与经济学家Epstein在Econometrica发表的论文，建立了被称为“Chen-Epstein”的资产定价模型，该文发展了诺贝尔经济学奖得主Lucas的资产定价理论，是中国大陆学者第一次在该国际权威杂志上发表的论文。得到包括诺贝尔经济学奖得主Sargent和Hansen以及美国联邦储备银行等一批著名学者和金融业界人士的多次引用和重要评价。

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                                            　数学与统计学院

2023年06月15日