State polynomials: positivity, optimization and nonlinear Bell inequalities

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 d...

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Bibliographic Details
Published in:Mathematical programming Vol. 207; no. 1-2; pp. 645 - 691
Main Authors: Klep, Igor, Magron, Victor, Volčič, Jurij, Wang, Jie
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024
Springer
Springer Verlag
Subjects:
Algebra
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Functional Analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization and Control
Physics
Quantum Physics
Theoretical
Semidefinite programming
Hilbert’s 17th problem
State optimization
State polynomial
Network scenario
13J30
81-08
46L30
46N50
Noncommutative polynomial
Polynomial Bell inequality
Positivstellensatz
90C22
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-02024-5