Optimization Over Trace Polynomials

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, an...

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Bibliographic Details
Published in:Annales Henri Poincaré Vol. 23; no. 1; pp. 67 - 100
Main Authors: Klep, Igor, Magron, Victor, Volčič, Jurij
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.01.2022
Springer Nature B.V
Springer Verlag
Subjects:
Bell's inequality
Classical and Quantum Gravitation
Convergence
Dynamical Systems and Ergodic Theory
Elementary Particles
Functional Analysis
Information theory
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical Physics
Mathematics
Operators (mathematics)
Optimization
Optimization and Control
Physics
Physics and Astronomy
Polynomials
Quantum Field Theory
Quantum phenomena
Quantum Physics
Relativity Theory
Theoretical
81-08
(Pure) trace polynomial
46N50
Semidefinite programming
Von Neumann algebra
Noncommutative polynomial
Trace optimization
Positivstellensatz
90C22
Semialgebraic set
47N10
13J30
ISSN:1424-0637, 1424-0661
Online Access:Get full text
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Summary: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.
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ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-021-01095-4