State polynomials: positivity, optimization and nonlinear Bell inequalities

TL;DR

This paper develops a framework for optimizing polynomials in noncommuting variables over quantum states, with applications to quantum network correlations and Bell inequalities, introducing new Positivstellensatz results and hierarchies of relaxations.

Contribution

It introduces state polynomials and proves a Positivstellensatz analogue, along with a hierarchy of semidefinite relaxations for quantum polynomial optimization, extending classical polynomial optimization methods to the quantum setting.

Findings

State polynomials positive over all matrices are sums of squares with denominators.

A state polynomial Positivstellensatz fails in general, unlike classical cases.

The hierarchy converges monotonically to the quantum optimization problem's optimum.

Abstract

This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also established that a Krivine-Stengle Positivstellensatz fails to hold in the state polynomial setting. Further, archimedean Positivstellens\"atze in the spirit of Putinar and Helton-McCullough are 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 Navascu\'es-Pironio-Ac\'in 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 and the bilocal tripartite scenario are analyzed. To reduce the size of the constructed SDPs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited. To obtain the above-mentioned results, techniques from noncommutative algebra, real algebraic geometry, operator theory, and convex optimization are employed.