Positivity of state, trace, and moment polynomials, and applications in quantum information
Felix Huber, Victor Magron, Jurij Volčič
TL;DR
This work unifies state, trace, and moment polynomials as noncommutative polynomials augmented by positive functionals and develops two main Positivstellensätze—Hilbert–Artsin for unconstrained and Archimedean for bounded problems—to certify positivity. It then builds convergent semidefinite programming hierarchies, with primal/dual formulations expressed via state Hankel matrices and localizing matrices, that converge to the optimal value under archimedean assumptions. The framework is applied to key quantum-information problems, including dimension-free entanglement witnesses for Werner states, nonlinear Bell inequalities in networks, quantum uncertainty relations via Lovász-theta-based relaxations, and SDP-based bounds for quantum codes. The approach provides a computationally tractable, algebraically unified toolkit linking operator-algebra, invariant theory, and real algebraic geometry to practical quantum-information questions, with potential extensions to trace and moment settings through appropriate constraints. Overall, the paper delivers both foundational theory and actionable SDP-based methods for optimizing noncommutative polynomials in quantum contexts.
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). While 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.