Optimization Over Trace Polynomials

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity subject to constraints is presented, a hierarchy semidefinite relaxations converging monotonically the optimum polynomial tracial provided. This can be seen as analog Pironio, Navascu\'es Ac\'in scheme [New J. Phys., 2008] for optimization noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction applied extract optimizers problem if flatness extremality conditions are satisfied. These sufficient obtain finite convergence our hierarchy. results obtained violations Bell inequalities theory. main techniques used paper inspired real algebraic geometry, operator algebra.