Tensor Products of Operator Systems and the Kadison-Singer Problem

在这个项目中，首席研究员将研究两个主要主题：(1)Kadison-Singer问题和(2)算子系统的张量积理论。卡迪森-辛格问题是这个数学领域的一个重大问题，自1954年以来一直没有解决。首席调查员最近在这个问题上取得了一些进展，并将探索三种新的攻击途径。算子系统和完全正映射在多个数学领域中发挥着核心作用，包括量子计算、量子信息论以及在C*-代数和von Neumann代数中的应用。首席研究人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。被称为框架理论的数学领域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，例如对声波采样，将其刻录到CD上，然后从CD回放音乐。工程师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。卡迪森-辛格问题的进展应该会转化为对冗余如何表现的比目前存在的更准确的理解。粗略地说，它问的是，具有“有限冗余”的系统是否总是可以划分为有限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。尽管没有人能预测量子计算机是否会被建造出来，但如果真的建成了，开发出足够的人力资源，在使用它们时具有竞争力，肯定对国家利益至关重要。因此，主要研究人员的学生被介绍到这个领域，他在算子系统张量理论方面的工作应用于量子环境中的平行结构问题。

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.