State polynomial optimization for nonlinear Bell inequalities

This talk focuses on optimization over state polynomials, i.e., polynomials in noncommuting variables and formal states of their products.

An archimedean Positivstellensatz in the spirit of Putinar and Helton-McCullough is presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín scheme for optimization of noncommutative polynomials.

The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite scenario are analyzed.

To reduce the size of the semidefinite programs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited.

This is based on the joint work with Igor Klep, Jurij Volčič, and Jie Wang.