By investiating the construction of quantum isomorphic, non-isomorphic graphs from binary constraint system games of Atserias et al, David Roberson and I showed that for every homogeneous solution group, there is a graph whose quantum automorphism group is isomorphic to the discrete dual of the solution group. This allowed us to construct a first example of a graph that has quantum symmetry but finite quantum automorphism group. It furthermore led to a first example of a pair of two quantum isomorphic, non-isomorphic graphs that both do not have quantum symmetry.
Then I worked on finding a new pair of quantum isomorphic, non-isomorphic graphs. Studying the E8 root system and the action of Pauli matrices on it, I was able to construct such a pair. The graphs involved are additionally strongly regular, making them the first known pair of strongly regular quantum isomorphic, non-isomorphic graphs. Moreover, I proved that using Godsil-McKay switching on the pair, one obtains more quantum isomorphic, non-isomorphic graph.
Furthermore, I worked on self-testing. Together with Laura Mančinska, I constructed the first example of a nonlocal game that is a self-test which is not robust. For this, we introduced the „or“-game and used it together with a game constructed by Slofstra. Additionally, we found a nonlocal game that does not self-test any state, which was also not known before. We could once again use the „or“-game for constructing the example. Kochen-Specker sets played an important role for finding the appropriate game. Another task in self-testing was to lift the common assumption that quantum strategies consist of projective measurements of a pure quantum state. In the absence of any prior knowledge we should model these quantum strategies as measuring a mixed state using positive, operator valued measurements. We establish a theorem allowing us to promote most existing self-testing results to their assumption-free variants. This was done together with Pedro Baptista, Ranyiliu Chen, Jed Kaniewski, David Lolck, Laura Mančinska and Thor Nielsen.
Quantum magic squares are generalizations of quantum permutation matrices, a concept that is central to the theory of quantum automorphism groups of graphs. Together with Andreas Bluhm and Ion Nechita, I investigated those from the viewpoint of polytope compatibility. We proved that semi-classical quantum magic squares correspond to elements in the minimal matrix convex set of the Birkhoff polytope, whereas general quantum magic squares are in correspondence with elements in the maximal one.
Together with David Roberson, I found that several Hadamard graphs have quantum symmetry and exactly one non-trivial automorphism. This does not settle the question of finding a graph with quantum symmetry and trivial automorphism group, but gives us evidence that such a graph exists.
