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.

在这个项目中，主要研究者将追求两个主要主题：（1）Kadison-Singer问题和（2）算子系统的张量积理论。Kadison-Singer问题是这一数学领域的一个重大问题，自1954年以来一直没有解决。该问题的主要研究者最近取得了一些进展，并将探索三种新的攻击途径。算子系统和完全正映射在数学的几个领域中发挥着核心作用，包括量子计算，量子信息论，以及C*-代数和冯诺依曼代数的应用。首席研究员将继续发展算子系统的一般张量理论，并继续将该理论应用于量子信息和量子计算中的问题。数学领域被称为框架理论，涉及用于对各种类型的信号进行采样，然后从样本中重建信号的系统，例如当对声波进行采样时，将其刻录到CD上，然后播放CD中的音乐。工程师们总是在这样的系统中建立冗余（或过采样），以改善样本数值误差的影响。在Kadison-Singer问题上的进展应该转化为对冗余行为的更精确的理解。粗略地说，它问的是，具有“有限冗余”的系统是否总是可以被划分为多个没有冗余的系统。该项目的第二个目标是发展量子信息理论和量子计算的数学。虽然没有人能预测量子计算机是否会被建造出来，但如果是的话，开发足够的人力资源，使其在使用上具有竞争力，对国家利益来说肯定是至关重要的。因此，首席研究员的学生介绍了这一领域，他的工作张量理论的运营商系统的应用问题的并行结构的量子设置。