Periodic Reporting for period 1 - NONLOCQUANT (Nonlocality in quantum groups)
Berichtszeitraum: 2021-06-01 bis 2023-05-31
Quantum symmetries of finite spaces were first characterized by Wang in 1997 using compact quantum groups. This led to the definition of the quantum symmetric group. Motivated by this, Banica and Bichon introduced quantum automorphism groups of finite graphs, capturing their quantum symmetries. Recently, connections to the isomorphism game, a nonlocal game constructed by Atserias et al., were drawn. The project „Nonlocality in quantum groups“ was to further investigate those connections.
The first objective was to study quantum symmetries and nonlocal symmetries of graphs. Nonlocal symmetries are quantum symmetries that can be observed by players of the associated isomorphism game. The goals include finding new quantum isomorphic, non-isomorphic graphs and a graph with quantum symmetry and trivial automorphism group.
The second objective focused on self-testing. In nonlocal games, self-testing results guarantee that optimal quantum strategies of a nonlocal game are unique, in some sense.
In a third objective, we wanted to use quantum information theoretic tools for quantum groups, to further foster exchange between the areas.
The first objective was to study quantum symmetries and nonlocal symmetries of graphs. Nonlocal symmetries are quantum symmetries that can be observed by players of the associated isomorphism game. The goals include finding new quantum isomorphic, non-isomorphic graphs and a graph with quantum symmetry and trivial automorphism group.
The second objective focused on self-testing. In nonlocal games, self-testing results guarantee that optimal quantum strategies of a nonlocal game are unique, in some sense.
In a third objective, we wanted to use quantum information theoretic tools for quantum groups, to further foster exchange between the areas.
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.
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.
The research carried out in the project led to several new developments in the theory of quantum automorphism groups of graphs and self-testing, for example group C^*-algebras realizable as quantum automorphism groups of graphs, new pairs of quantum isomorphic graphs and a non-robust self-test. All results are freely available on arXiv.org and thus the scientific papers produced during the fellowship will be easily accessible, allowing further investigation on the topic.