Quantized Symmetries in Operator Algebras and Quantum Information

世纪以来，人们都知道我们的物理宇宙是由量子力学控制的。 量子力学的一个基本特征是，如果你对一个物理系统进行测量，它会永久地改变系统的状态。 这意味着对某些物理系统进行测量的顺序很重要。 在这种情况下，我们说测量是“非交换的”。 量子力学固有的非交换性由算子代数理论描述，算子代数是数学的分支，研究如何有效地操纵和编码任意大的数值数据矩阵。 今天，算子代数和量子理论都是数学科学的成熟和独立的分支。然而，近年来，随着量子计算和量子信息科学的出现，算子代数的更多数学理论和量子力学的更多物理理论之间的富有成效的相互作用再次出现。 该项目的主要目标是探索和深化算子代数理论和量子信息理论（QIT）之间的新兴联系。 在这个项目中的中心研究对象是量化对称算子代数和QIT。 理解系统的对称性长期以来一直是数学和物理问题的有力工具，而算子代数和QIT的高度非交换性质自然导致发现更灵活的对称性概念，称为量子对称性。 在这个项目中，我们将使用量子对称性的各种数学体现来提供算子代数理论和QIT之间的联系，我们将使用这种联系来提供对这两个重要领域的新见解。本项目将通过培养研究生和本科生为美国劳动力的发展做出贡献。本研究项目分为两个主要方向。第一个方向涉及的结构理论的一些新的和有趣的类的算子代数所产生的量子同构空间。 量子同构空间在量子隐形传态和量子量子信息技术中的超密集编码方案的研究中以及量子置换群的表示理论中自然出现。 这些代数的研究使我们提出了一个量子模拟Lueck的行列式猜想从几何群论和调查其潜在的应用，以建设新的例子强1-有界和强固体冯诺依曼代数。该项目的第二个方向涉及首席研究员对量子对称性和QIT之间相互作用的开创性工作。在这里，几何和代表性理论的工具将被用来寻找应用程序的非本地游戏中产生的QIT建设的新的例子超线性离散quantum groups.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

For over a century it has been known that our physical universe is governed by quantum mechanics. A fundamental feature of quantum mechanics is that if you perform a measurement on a physical system, it has the effect of permanently altering the state of the system. This means that the order in which one performs measurements on certain physical systems matters. In this context, we say that measurement is "non-commutative". The inherent non-commutativity of quantum mechanics is described by the theory of operator algebras, which is the branch of mathematics that investigates how to efficiently manipulate and encode arbitrarily large matrices of numerical data. Today, operator algebras and quantum theory are both well-established and independent branches of the mathematical sciences. However, in recent years, with the advent of quantum computation and quantum information science, there has been a resurgence of fruitful interactions between the more mathematical theory of operator algebras and the more physical theory of quantum mechanics. The primary goal of this project is to explore and deepen the emerging connections between operator algebra theory and quantum information theory (QIT). The central objects of study in this project are quantized symmetries in operator algebras and QIT. Understanding the symmetries of systems has long been a powerful tool in mathematical and physical problems, and the highly non-commutative nature of both operator algebras and QIT naturally leads to the discovery of more flexible notions of symmetries, called quantum symmetries. In this project, we will use various mathematical incarnations of quantum symmetries to provide a link between operator algebra theory and QIT, and we will use this link to provide new insights into both of these important fields. This project will contribute to the development of the US workforce through the training of graduate and undergraduate students.This research project breaks up into two main directions. The first direction concerns the structural theory of some new and interesting classes of operator algebras arising from quantum isomorphism spaces. Quantum isomorphism spaces appear naturally in the study of quantum teleportation and super-dense coding schemes in QIT, and also in the representation theory of quantum permutation groups. The study of these algebras leads us to propose a quantum analogue of Lueck’s determinant conjecture from geometric group theory and investigate its potential applications to constructing new examples of strongly 1-bounded and strongly solid von Neumann algebras. The second direction of this project relates to the Principal Investigator's pioneering work on the interactions between quantum symmetries and QIT. Here, geometric and representation-theoretic tools will be used to find applications of non-local games arising in QIT to the construction of new examples of hyperlinear discrete quantum groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.