Abstract

Quantum isomorphisms of graphs form a bridge between between noncommutative geometry (NCG) and quantum information theory (QIT), as they connect quantum automorphism groups of graphs with nonlocal games. This makes it possible to use techniques from QIT in NCG and vice versa. In this talk, I will present a striking application of this connection, where we use ideas from QIT to prove a surprising result in NCG . Using a construction similar to the Mermin–Peres magic square from QIT , we construct graphs with trivial automorphism group and non-trivial quantum automorphism group, which shows that even graphs with no symmetry at all can have hidden quantum symmetries. To our knowledge, these are the first known examples of any kind of commutative spaces in NCG with this property. This talk is based on joint work with David E. Roberson (Technical University of Denmark) and Simon Schmidt (Ruhr University Bochum).