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？tze 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 s–Pironio–Ac n 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.  

　　Publication:  

　　Mathematical Programming, 03 November 2023  

　　http://dx.doi.org/10.1007/s10107-023-02024-5  

　　Author:  

　　Igor Klep  

　　Department of Mathematics, Institute of Mathematics, Physics and Mechanics, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia  

　　Victor Magron  

　　LAAS-CNRS, Institute of Mathematics from Toulouse, Toulouse, France  

　　Jurij Vol？i？  

　　Department of Mathematics, Drexel University, Philadelphia, PA, USA  

　　Jie Wang  

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China  

　　Email: wangjie212@amss.ac.cn