KMS States of Quantum Cuntz-Krieger Algebras

C*-代数理论起源于20世纪30年代的量子力学研究，现在是现代数学分析的重要组成部分，在数学科学中有着广泛的应用。C*-代数自然而然地与各种感兴趣的数学对象有关，包括群、动力系统和离散图。本课题主要研究与量子图有关的C*-代数的结构和性质。作为离散图经典概念的相对较新的推广，量子图已被证明在量子信息论中很有用：正如经典离散图编码由于经典通信信道中的噪声而造成的混乱一样，量子图编码由于量子信道中的噪声而造成的混乱。该项目将产生新的方法来分析量子Cuntz-Krieger代数及其基本量子图的结构，并探索它们与量子信息理论的相互作用，这是一个日益引起全球兴趣的话题。本科生的教育机会将通过研究项目和PI所在机构的量子信息入门理论的新的跨学科认证计划来提供。学生研究人员和访问演讲者将被招募，重点是多样性和代表性。给定一个简单的离散图，该图的Cuntz-Krieger代数是编码该图的边关系的通用C*-代数。C*-代数上的Kubo-Martin-Schwinger(KMS)态可以物理地解释为量子系统的热平衡态。2003年，Exel利用Cuntz-Krieger代数和图的Exel交叉积之间的同构对简单离散图的Cuntz-Krieger代数上的KMS状态进行了分类，交叉积是一种通用的C*-代数，它编码了图的无限路径空间上的自然动态。对于量子图，2021年定义了与其Cuntz-Krieger代数类似的量子Cuntz-Krieger代数。这位主要研究人员和她的合作者已经为某些类型的量子图构造了Exel交叉积，并证明了这些Exel交叉积与相应的量子Cuntz-Krieger代数的商同构。这个项目的第一个主要目标是设计任意量子图的Exel交叉积的规范结构，并研究它与相应的量子Cuntz-Krieger代数的关系。这个项目的第二个主要目标是对量子图的Exel交叉积上的KMS状态进行分类，并遵循Exel在经典设置中的技术，使用第一个目标中建立的关系来对相关的量子Cuntz-Krieger代数上的KMS状态进行分类。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。

The theory of C*-algebras, which originated in the 1930s in the study of quantum mechanics, is now a vital part of modern mathematical analysis, with applications across the mathematical sciences. C*-algebras arise naturally in connection with a variety of mathematical objects of interest, including groups, dynamical systems, and discrete graphs. This project concerns the structure and properties of C*-algebras associated to quantum graphs. A relatively recent generalization of the classical notion of a discrete graph, quantum graphs have proven to be useful in quantum information theory: just as classical discrete graphs encode confusion due to noise in a classical communication channel, quantum graphs encode confusion due to noise in a quantum channel. The project will generate new methods for analyzing the structure of quantum Cuntz-Krieger algebras and their underlying quantum graphs, and explore their interplay with quantum information theory, a topic of growing global interest. Educational opportunities for undergraduates will be provided through research projects, and a new, interdisciplinary certification program in introductory quantum information theory at the PI’s home institution. Student researchers and visiting speakers will be recruited with a focus on diversity and representation.Given a simple discrete graph, the Cuntz–Krieger algebra for the graph is a universal C*-algebra which encodes the graph’s edge relations. The Kubo-Martin-Schwinger (KMS) states on a C*-algebra can be physically interpreted as states of thermal equilibrium for a quantum system. The KMS states on the Cuntz–Krieger algebra of a simple discrete graph were classified by Exel in 2003 using an isomorphism between the Cuntz–Krieger algebra and the graph’s Exel crossed product, which is a universal C*-algebra that encodes natural dynamics on the graph’s infinite path space. For a quantum graph, an analogue of its Cuntz–Krieger algebra, called a quantum Cuntz–Krieger algebra, was defined in 2021. The principal investigator and her collaborators have since constructed Exel crossed products for some classes of quantum graphs and shown these Exel crossed products to be isomorphic to a quotient of the corresponding quantum Cuntz–Krieger algebras. The first major objective of this project is to design a canonical construction of an Exel crossed product for an arbitrary quantum graph and study its relationship to the corresponding quantum Cuntz–Krieger algebra. The second major objective of this project is to classify the KMS states on the Exel crossed product for a quantum graph and, following Exel’s techniques in the classical setting, use this relationship established in the first objective to classify the KMS states on the associated quantum Cuntz–Krieger algebra.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.