Quantum Perspectives in Banach and Metric Spaces

在许多有趣的情况下，人们有一种定量的方式来表达两个物体之间的距离。一个基本的例子是空间中位置之间的物理距离，一个现代的例子是测量两个单词的相似程度（这是软件自动纠正我们输入的方式）。用来概括这个思想的数学概念是度量空间。特别有趣的类型的空间是那些来自网络,也就是说,这些测量的两个节点之间的距离是由路径连接的最佳选择使用网络中的链接:俄克拉荷马城和波士顿之间没有直达航班,所以用于旅行重要的不是这两个城市之间的地理距离,而是如何从一个到另一个使用可用的商业航班。这种网络类型的空间在信息论中扮演着重要的角色，信息论是通信的数学框架，其中信息被编码成一串0和1，就像今天的计算机一样。该项目研究由量子信息理论产生的相应量子网络类型空间的形状，量子信息理论对通信系统进行建模，其中信息现在以量子力学系统的状态编码。更好地了解这些量子空间的形状对理解量子过程的能力具有重要意义，这些量子过程将被未来的量子计算机所使用。此外，通过外联、博士后指导和参与本科生研究，该项目将有助于STEM领域学生和研究人员的成长和多样化。该项目将重点关注由各种量化模型引起的几何问题。它分为三个部分。第一部分是关于量子图和其他量子度量空间，重点是量子扩展器和量子度量空间大尺度几何理论的进一步发展。第二部分提出利用量子信息论工具研究非交换序列和函数空间的非线性几何。重点是研究Banach空间几何、算子代数理论、几何群理论和理论计算机科学中感兴趣的几种广义Mazur映射的非交换版本，以及加权非交换Lebesgue p空间的新推广的插值性质。第三部分主要讨论算子空间，算子空间是量子巴拿赫空间的一种。研究它们的非线性几何是一个很有意义的课题。该项目打算利用来自张量积线性理论的工具，继续发展算子空间的非线性几何理论，并研究算子空间的渐近张量积（其巴拿赫空间的对应项最近在量子信息理论中得到了应用）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In many situations of interest, one has a quantitative way of expressing how far apart two objects are. A basic example is physical distance between locations in space, and a modern one is to measure how similar two words are (which is how software autocorrects our typing). The mathematical notion used to encapsulate this idea is that of a metric space. Particularly interesting kinds of such spaces are those that arise from networks, that is, those for which the measure of distance between two nodes is given by the optimal choice of a path connecting them using the links in the network: there are no direct flights between Oklahoma City and Boston, so for travel purposes what matters is not the geographical distance between the two cities but rather how to get from one to the other using the available commercial flights. Such network-type spaces play an important role in Information Theory, the mathematical framework for communications where information is encoded in a string of 0’s and 1’s as in today’s computers. The project studies the shape of the corresponding quantum network-type spaces that arise from Quantum Information Theory, which models communication systems where information is now encoded in the state of a quantum-mechanical system. Better knowledge of the shape of these quantum spaces has implications for the understanding of the capabilities of the quantum processes that will be used by the quantum computers of the future. In addition, through outreach, mentoring of postdocs, and involvement in undergraduate research, the project will contribute to the growth and diversification of the body of students and researchers in STEM fields. The project will focus on geometric questions arising from various models of quantization. It is divided into three parts. The first is about quantum graphs and other quantum metric spaces, with an emphasis on quantum expanders and the further development of a theory of the large-scale geometry of quantum metric spaces. The second part proposes to use Quantum Information Theory tools to study the nonlinear geometry of noncommutative sequence and function spaces. The focus is to study noncommutative versions of several generalized Mazur maps of interest in Banach space geometry, Operator Algebra Theory, Geometric Group Theory, and Theoretical Computer Science, as well as interpolation properties of a new generalization of weighted noncommutative Lebesgue p spaces. The third part is focused on operator spaces, which are a type of quantum Banach spaces. The study of their nonlinear geometry is a timely subject. The project intends to adapt tools coming from the linear theory of tensor products to continue the development of a nonlinear geometric theory for operator spaces, and to study asymptotic tensor products for operator spaces (whose Banach space counterparts have in recent times found applications in Quantum Information Theory).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.