Quantum Perspectives in Banach and Metric Spaces

在许多感兴趣的情况下，人们有一种定量的方式来表示两个对象之间的距离。一个基本的例子是空间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(这就是软件如何自动纠正我们的打字)。用来概括这一概念的数学概念是度量空间。特别有趣的这类空间是由网络产生的，也就是说，两个节点之间的距离的测量是通过使用网络中的链路连接它们的最佳路径来给出的：俄克拉荷马城和波士顿之间没有直达航班，所以对于旅行来说，重要的不是两个城市之间的地理距离，而是如何使用可用的商业航班从一个城市到另一个城市。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，在信息论中，信息被编码成0‘S和1’S的字符串，就像今天的计算机一样。该项目研究产生于量子信息论的相应量子网络型空间的形状，量子信息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进行编码。更好地了解这些量子空间的形状，对于理解未来量子计算机将使用的量子过程的能力具有重要意义。此外，通过外展、指导博士后和参与本科生研究，该项目将有助于STEM领域的学生和研究人员的成长和多样化。该项目将专注于由各种量子化模型引起的几何问题。全文共分为三个部分。第一个是关于量子图和其他量子度量空间，重点是量子展开器和量子度量空间的大规模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对易序列和函数空间的非线性几何。重点研究了Banach空间几何、算子代数理论、几何群论和理论计算机科学中几个感兴趣的广义Mazur映射的非对易形式，以及加权非对易Lebesguep空间的一个新推广的内插性质。第三部分主要研究了量子Banach空间中的算子空间。对它们的非线性几何的研究是一个适时的课题。该项目旨在采用来自张量积的线性理论的工具来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Banach空间对应的对象最近在量子信息论中找到了应用)。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.