The singular bivariate quartic tracial moment problem

TL;DR

This paper investigates the singular bivariate quartic tracial moment problem, providing complete solutions for low-rank cases, exploring the existence and uniqueness of measures, and analyzing the role of flat extensions.

Contribution

It extends the classical moment problem to the tracial setting for singular matrices, offering explicit solutions and insights into measure existence, uniqueness, and the limitations of flat extensions.

Findings

Complete solution for rank ≤ 5 cases.

Reduction of rank 6 cases to linear matrix inequalities.

Flat extensions are not necessary for measure existence in the tracial case.

Abstract

The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure $\mu$ on $\mathbb R^n$. In \cite{BK12} Burgdorf and Klep introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of $\mu$ with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite $7\times 7$ moment matrix $\mathcal M_2$ can be represented with tracial moments \cite{BK10,BK12}. In this article the case of singular $\mathcal M_2$ is studied. For $\mathcal M_2$ of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For $\mathcal M_2$ of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix $\mathcal M_2$ is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case.