The singular bivariate quartic tracial moment problem

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.