Quantum Graphs: Different Perspectives

Abstract

In this thesis, we investigate two different notions of quantum graphs. The first approach is through quantum adjacency matrices, while, the second approach is through bimodules over finite-dimensional algebras. We establish the equivalence between these approaches, following the work by M. Daws [Daw24]. Along the way, we explore the role of quantum graphs as operator systems within the context of quantum information theory, serving as an extension of confusability graphs in classical information theory. Furthermore, we explore the concept of quantum isomorphism between quantum graphs. We use the standard definition of quantum isomorphism, defined via the quantum adjacency matrix approach, to introduce an equivalent notion in the bimodules approach (employing the equivalence between the approaches). The goal of this work is to present and contribute to the growing knowledge of quantum graphs and isomorphisms between them.