Tensor Products of Operator Systems and the Kadison-Singer Problem

算子系统的张量积和 Kadison-Singer 问题

• 批准号: 1101231
• 负责人: Vern Paulsen
• 金额: $ 21.14万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101231&HistoricalAwards=false
• 关键词: Tensor; Products; Operator; Systems; Kadison

In this project, the principal investigator will pursue two major themes: (1) the Kadison-Singer problem and (2) the theory of tensor products of operator systems. The Kadison-Singer problem is a major problem in this area of mathematics that has been unsolved since 1954. The principal investigator has made some recent progress on the problem and will explore three new avenues of attack on it. Operator systems and completely positive maps play a central role in several areas of mathematics, including quantum computing, quantum information theory, and applications to C*-algebras and von Neumann algebras. The principal investigator will continue to develop the general tensor theory of operator systems and continue to apply this theory to problems in quantum information and quantum computing.The area of mathematics known as frame theory is concerned with systems that are used to sample signals of various types and then to reconstruct the signals from the samples, such as one does when sampling a soundwave, burning it to a CD, then playing back the music from the CD. Engineers always build redundancy (or oversampling) into such systems in order to ameliorate the effects of errors in the numerical values of the samples. Progress on the Kadison-Singer problem should translate to a more precise understanding of how redundancy behaves than exists at the present time. Roughly, it asks whether or not systems with "finite redundancy" can always be divided into finitely many systems with no redundancy. The second goal of the project is concerned with developing the mathematics of quantum information theory and quantum computing. Although no one can predict whether or not quantum computers will ever be built, if they are, it is certainly vital to the national interests to have developed sufficient human resources to be competitive in putting them to use. Consequently, the principal investigator's students are introduced to this area, and his work on the tensor theory of operator systems has applications to questions about parallel structure in the quantum setting.

在该项目中，主要研究人员将遵循两个主要主题：（1）Kadison-Singer问题和（2）操作员系统的张量产品理论。 Kadison-Singer问题是自1954年以来无法解决的这一数学领域的主要问题。首席研究人员在该问题上取得了一些最新进展，并将探索三个新的攻击途径。操作员系统和完全正面的地图在数学的多个领域中起着核心作用，包括量子计算，量子信息理论以及对C*-Algebras和von Neumann代数的应用。 The principal investigator will continue to develop the general tensor theory of operator systems and continue to apply this theory to problems in quantum information and quantum computing.The area of​​ mathematics known as frame theory is concerned with systems that are used to sample signals of various types and then to reconstruct the signals from the samples, such as one does when sampling a soundwave, burning it to a CD, then playing back the music from the CD.工程师始终将冗余（或过度采样）构建到此类系统中，以减轻样品数值中错误的影响。 Kadison-Singer问题的进展应该转化为对冗余的行为的更精确的理解，而不是目前的情况。粗略地，它询问是否始终将具有“有限冗余”的系统分为有限的许多系统，没有冗余。该项目的第二个目标与开发量子信息理论和量子计算的数学有关。尽管没有人能够预测是否会建造量子计算机，但如果有的话，对于国家利益来说，开发足够的人力资源以使其使用它们的使用肯定至关重要。因此，将主要研究人员的学生介绍给该领域，他在张量操作者系统方面的工作将有关量子环境中并行结构的问题应用于量子。