Positivity of State, Trace, and Moment Polynomials and Applications in Quantum Information

Felix Huber (), Victor Magron () and Jurij Volčič ()
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Felix Huber: University of Gdańsk, Faculty of Mathematics, Physics and Informatics, Institute of Theoretical Physics and Astrophysics
Victor Magron: LAAS CNRS
Jurij Volčič: University of Auckland, Department of Mathematics

Chapter 89 in Operator Theory, 2026, pp 2773-2802 from Springer

Abstract: Abstract State, trace, and moment polynomials are polynomial expressions in several operator or random variables and positive functionals on their products (states, traces, or expectations). These concepts, and in particular their positivity and optimization, arose from problems in quantum information theory, yet they naturally fit under the umbrella of multivariate operator theory. This survey presents state, trace, and moment polynomials in a concise and unified way and highlights their similarities and differences. The focal point is their positivity and optimization. Sums of squares certificates for unconstrained and constrained positivity (Positivstellensätze) are given, and parallels with their commutative and freely noncommutative analogs are discussed. They are used to design a convergent hierarchy of semidefinite programs for optimization of state, trace, and moment polynomials. Finally, circling back to the original motivation behind the derived theory, multiple applications in quantum information theory are outlined.

Keywords: Noncommutative polynomial optimization; State polynomial; Trace polynomial; Moment polynomial; Positivstellensatz; Hierarchy of semidefinite programs; Nonlinear Bell inequalities; Entanglement detection; Uncertainty relations; Quantum codes (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_88

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