Abstract

Quantum graphs are a non-commutative generalization of classical graphs that have received significant attention in recent years due to their close connections to operator spaces, C*-algebras, quantum information theory and quantum groups. In this talk, I will provide an introduction to quantum graphs and discuss some quantum-input and quantum-output nonlocal games involving quantum graphs.    A nonlocal game typically involves two non-communicating players who jointly try to convince a verifier about something by answering with a winning pair of responses to questions posed by the verifier. In this talk, I will survey some recent approaches to nonlocal homomorphism games for quantum graphs and discuss how these notions connect with one another. In the case of classical graphs, it is well-known that winning strategies for the graph isomorphism game arise from certain quantum permutation matrices that intertwine the adjacency matrices of the two graphs. It will be highlighted that analogous results also hold true in the setting of quantum graphs, where the quantum permutation matrix may be replaced by more general non-commutative permutation matrices over quantum sets intertwining the respective quantum graph structures.