There is a plethora of prior research in quantum information theory involving reformulating problems as optimization of noncommutative polynomials.
One famous application is to characterize the set of quantum correlations and to investigate entanglement.
Noncommutative analogs of the Lasserre's hierarchy have been introduced by experts from real algebraic geometry (Helton and McCullough) and from quantum physics (Navascues, Pirionio, Acin).
So far, prior research in quantum information theory focused intensively on reformulating problems as eigenvalue optimization of noncommutative polynomials.
Motivated by certain quantum information problems, including polynomial Bell inequalities and Werner witnesses, the team is also interested in deriving hierarchies to optimize over trace polynomials, i.e., polynomials in noncommuting variables and traces of their products, as well as noncommutative analogs of Christoffel-Darboux kernels.