TY - JOUR

T1 - State polynomials

T2 - positivity, optimization and nonlinear Bell inequalities

AU - Klep, Igor

AU - Magron, Victor

AU - Volčič, Jurij

AU - Wang, Jie

N1 - Publisher Copyright:
© 2023, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - 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 Positivstellensätze 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 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 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.

AB - 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 Positivstellensätze 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 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 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.

KW - Hilbert’s 17th problem

KW - Network scenario

KW - Noncommutative polynomial

KW - Polynomial Bell inequality

KW - Positivstellensatz

KW - Semidefinite programming

KW - State optimization

KW - State polynomial

UR - http://www.scopus.com/inward/record.url?scp=85175611695&partnerID=8YFLogxK

U2 - 10.1007/s10107-023-02024-5

DO - 10.1007/s10107-023-02024-5

M3 - Article

AN - SCOPUS:85175611695

SN - 0025-5610

JO - Mathematical Programming

JF - Mathematical Programming

ER -