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 main techniques used are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

Exploiting term sparsity in Noncommutative Polynomial Optimization

“…Recently a moment-SOHS hierarchy for optimization problems involving trace polynomials was proposed in [22]. It would be worth extending further our sparsityexploiting framework to handle trace polynomials.…”

“…In the same spirit, one can also obtain a hierarchy of SDP relaxations to approximate as closely as desired the minimal trace of an nc polynomial over an nc semialgebraic set [8,11,33]. We also refer the interested reader to [22] for the case of more general trace polynomials, i.e., polynomials in noncommutating variables and traces of their products.…”

Noncommutative Christoffel-Darboux Kernels

“…Sparsity exploiting hierarchies [KMP21,WM20] allow one to reduce the associated computational burden. The case of more general trace polynomials has been investigated in [KMV20]. Algorithms similar to the one from [HL05] allow one to extract optimizers of eigenvalue or trace minimization problems; see, e.g., [PNA10], [AL12, Chapter 21], [BKP16, Theorem 1.69] and [BCKP13].…”

Positive univariate trace polynomials

“…Namely, the failure of Connes' embedding conjecture [JNVWY20] implies that there is a noncommutative polynomial whose trace is positive on all matrix contractions, but negative on a tuple of operator contractions from a tracial von Neumann algebra [KS08], which obstructs the existence of a clean trace-of-squares certificate for matrix positivity in general. There are however Positivstellensätze for positivity of multivariate trace polynomials on von Neumann algebras subject to archimedean constraints [KMV20]. In a different direction, tracial inequalities of analytic functions are heavily studied in relation to monotonicity, convexity and entropy in quantum statistical mechanics [Car10].…”