Optimization Over Trace Polynomials

Igor Klep, Victor Magron, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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 (Pironio et al. in New J. Phys. 10(7):073013, 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.

Original languageEnglish
Pages (from-to)67-100
Number of pages34
JournalAnnales Henri Poincare
Issue number1
StatePublished - 1 Jan 2022
Externally publishedYes


  • (Pure) trace polynomial
  • Noncommutative polynomial
  • Positivstellensatz
  • Semialgebraic set
  • Semidefinite programming
  • Trace optimization
  • Von Neumann algebra

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics


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