The tracial moment problem and trace-optimization of polynomials
In this paper the authors consider the trace optimization of polynomials in non-commuting (nc) variables. To obtain a lower bound on the trace of such polynomials a relaxation is formulated in terms of semi-definite programming (SDP), which is based on sums of hermitian squares and commutators. It is shown that the lower bound is attained if the dual solution of the SDP satisfies a certain condition called flatness. Furthermore, the authors provide a procedure to determine the global minimizer of the trace. The paper is very well written and completed by numerous helpful examples. After the introduction and basic notations (nc polynomials, sum of hermitian squares), which are given in Section 1, the cyclic equivalence relation for nc polynomials, which considers two polynomials to be equal if their difference is a commutator, is introduced in Section 2. Based on this relation the convex cone \(\theta\) of nc polynomials that are cyclic equivalent to a sum of hermitian squares is stated. Using the Gram-Matrix method it is shown that each nc polynomial \(f\) is within the cone \(\theta\) if and only if it is cyclic equivalent to a product \(W^\ast G W\), where \(G\) is is a positive semi-definite matrix, \(W\) is a vector of all possible nc polynomials with degree less than half of the degree of \(f\) and \(W^\ast\) the corresponding vector of reversed nc polynomials. In Section 3, the SDP relaxation of the trace optimization problem is presented. It is shown that the relaxation yields a lower bound of the original problem and satisfies strong duality. Furthermore, the truncated tracial moment problem is given. It generalizes the tracial moment problem of Bayer and Teichmann and is used to formulated the flatness condition that guarantees exactness for the solution of the relaxation and the original problem. The flatness condition for the solution of the dual SDP and the corresponding truncated tracial sequence \(y\) of degree \(2k\) requires that the Hankel matrix \(M_k(y)\) is flat over \(M_{k-1}(y)\), i.e., the ranks of \(M_k(y)\) and \(M_{k-1}(y)\) are identical. In the last section a Gelfand-Naimark-Segal construction and an Artin-Wedderburn block decomposition are used to compute a representation of a trace-minimizer for a numerical example.
scientific article; zbMATH DE number 6145980
scientific article