Files and links (1)
SubmittedarXiv.org - Non-exclusive license to distribute, Open
Journal article
State polynomials: positivity, optimization and nonlinear Bell inequalities
MATHEMATICAL PROGRAMMING
03 Nov 2023
Featured in Collection : UN Sustainable Development Goals @ Drexel
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 Positivstellensatze 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 Navascues-Pironio-Acin 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.
Metrics
Details
- Title
- State polynomials: positivity, optimization and nonlinear Bell inequalities
- Publication Details
- MATHEMATICAL PROGRAMMING
- Publisher
- SPRINGER HEIDELBERG; HEIDELBERG
- Grant note
- The authors thank Timotej Hrga for sharing his expertise and solving the large semidefinite program pertaining to one of the Bell inequality examples.
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Drexel University
- Web of Science ID
- WOS:001120174400001
- Scopus ID
- 2-s2.0-85175611695
- Other Identifier
- 991021860730204721
UN Sustainable Development Goals (SDGs)
This publication has contributed to the advancement of the following goals:
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Computer Science, Software Engineering
- Mathematics, Applied
- Operations Research & Management Science