Operator algebras between theory and application

理论与应用之间的算子代数

• 批准号: 1501103
• 负责人: Marius Junge
• 金额: $ 30万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-05-15 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501103&HistoricalAwards=false
• 关键词: Operator; algebras; between; theory; application

Order matters. The order in which certain operations are performed can dramatically change the outcome in real life and in the sciences. However, in the usual multiplication of numbers the order is irrelevant. Since the product AB is the same as BA one says that the factors A and B commute. Inspired by the fundamentals of quantum mechanics, mathematicians have investigated a new type of multiplication that respects the order of operations. During the last century this has led to spectacular new discoveries in mathematical theories embracing noncommutativity (i.e., allowing AB to be different from BA) such as noncommutative geometry or quantum (i.e., noncommutative) probability. In this line of research a similar program is applied to fundamental concepts in classical harmonic analysis such as Fourier series and estimates for solving differential equations in noncommutative spaces. Quite surprisingly, the abstract tools developed in this investigation are also useful in other disciplines. The research of the principal investigator will include a thorough analysis of quantum channels in quantum information theory. Research in quantum information theory usually takes place in computer science and physic departments. However, as long as quantum computers are not available in large numbers, the limitations and advantages of quantum computers can be understood only using theoretical, mathematical tools. The same applies for the capacities of devices transmitting information through the use of quantum mechanics. Interdisciplinary research in this work will also include mathematical aspects of big data and compressed sensing. All aspects of this research will also serve to enhance the teaching mission of the university, and in particular the formation of students who are familiar with pure mathematics and certain applications alike.The theory of operator algebras provides many important tools that are essential in understanding noncommutative aspects of classical objects, such as Brownian motion, derivatives and derivations, tangent and cotangent spaces, Laplace-Beltrami operators, singular integral kernels, quantum channels, and capacity of quantum channels. The project will aim to connect theoretical aspects of the theory of completely positive maps with more applied aspects in quantum information theory and harmonic analysis, in particular those analytic properties of operators that have a geometric or metric flavor. The proposed work on the Grothendieck program for triple-tensor norms belongs to the core subject in operator space theory but is also motivated by quantum information and compressed sensing. Previous research of the principal investigator related to quantum information theory has already demonstrated the potential to connect to topics in computer science and physics. The proposed new research on private capacity of channels may even have an impact beyond science.

订单很重要。执行某些操作的顺序可以极大地改变现实生活和科学的结果。但是，在通常的数字乘法中，顺序是无关紧要的。由于产品AB与BA相同，因此A和B上通勤的因素A和B上通勤。受量子力学的基本原理的启发，数学家研究了一种尊重运营顺序的新型乘法。在上个世纪，这导致了数学理论的壮观新发现（即允许AB与BA不同），例如非交流性几何形状或量子（即非交通性）概率。在这一研究中，类似的程序适用于经典谐波分析中的基本概念，例如傅立叶级数和解决非交通空间中微分方程的估计值。令人惊讶的是，本研究中开发的抽象工具在其他学科中也很有用。主要研究者的研究将包括对量子信息理论中量子通道的详尽分析。量子信息理论的研究通常发生在计算机科学和物理部门。但是，只要量子计算机没有大量可用，只有使用理论，数学工具才能理解量子计算机的局限性和优势。通过使用量子力学，设备的能力也适用于传输信息。这项工作中的跨学科研究还将包括大数据和压缩感测的数学方面。 All aspects of this research will also serve to enhance the teaching mission of the university, and in particular the formation of students who are familiar with pure mathematics and certain applications alike.The theory of operator algebras provides many important tools that are essential in understanding noncommutative aspects of classical objects, such as Brownian motion, derivatives and derivations, tangent and cotangent spaces, Laplace-Beltrami operators, singular integral kernels,量子通道和量子通道的能力。该项目将旨在将完全正面地图理论的理论方面与量子信息理论和谐波分析中的更多应用方面联系起来，尤其是那些具有几何或度量风味的操作员的分析性能。关于三张量规范的Grothendieck计划的拟议工作属于操作员空间理论中的核心主题，但也以量子信息和压缩感应为动机。与量子信息理论相关的主要研究者的先前研究已经证明了与计算机科学和物理学中的主题联系的潜力。拟议的关于渠道私人能力的新研究甚至可能会产生科学之外的影响。