Noncommutative polynomial optimization under symmetry
arXiv:2112.10803
Abstract
We present a general framework to exploit the symmetries present in the Navascu{é}s-Pironio-Ac{í}n semidefinite relaxations that approximate invariant noncommutative polynomial optimization problems. We put equal emphasis on the moment and sum-of-squares dual approaches, and provide a pedagogical and formal introduction to the Navascu{é}s-Pironio-Ac{í}n technique before working out the impact of symmetries present in the problem. Using our formalism, we compute analytical sum-of-square certificates for various Bell inequalities, and prove a long-standing conjecture about the exact maximal quantum violation of the CGLMP inequalities for dimension 3 and 4. We also apply our technique to the Sliwa inequalities in the Bell scenario with three parties with binary measurements settings/outcomes. Symmetry reduction is key to scale the applications of the NPA relaxation, and our formalism encompasses and generalizes the approaches found in the literature.
arXiv commentSupplementary material available on GitHub https://github.com/marieio/NPOsym