Optimization over trace polynomials

Abstract

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme [New J. Phys., 2008] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

32 pages

References in corpus (4)

• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Proper Analytic Free Maps
• Covariance Bell inequalities
• Positive maps and trace polynomials from the symmetric group

Cited by in corpus (8)

• Bell nonlocality in networks
• Dimension-free entanglement detection in multipartite Werner states
• Positive maps and trace polynomials from the symmetric group
• State polynomials: positivity, optimization and nonlinear Bell inequalities
• Positive univariate trace polynomials
• Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
• Positive maps from the walled Brauer algebra
• Exploiting term sparsity in Noncommutative Polynomial Optimization