KMS States of Quantum Cuntz-Krieger Algebras

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.