Sabine Burgdorf

The tracial moment problem and trace-optimization of polynomials

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace

On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings

{f(\underline {A})}

Optimization of Polynomials in Non-Commuting Variables

can attain for a tuple of matrices

On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings

{\underline {A}}

Detecting Sums of Hermitian Squares

? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix *-algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side—two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.

Selected Results from Algebra and Mathematical Optimization

Lieb and Seiringer stated in their reformulation of the Bessis–Moussa–Villani conjecture that all coefficients of the polynomial p(t) = tr((A +tr B) m ) are non-negative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m∈ℕ the coefficient of t 4 in p(t) is non-negative, using a connection to sums of Hermitian squares of non-commutative polynomials which has been established by Klep and Schweighofer. This implies by a well-known result of Hillar that the coefficients of t k are non-negative for 0 ≤ k ≤ 4, and by symmetry as well for m ≥ k ≥ m − 4.