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 μ on Rⁿ. Burgdorf and Klep (J Oper Theory 68:141–163, 2012) introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of μ 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 × 7 moment matrix M₂ can be represented with tracial moments (Burgdorf and Klep in C R Math Acad Sci Paris 348:721–726, 2010, 2012). In this article the case of singular M₂ is studied. For M₂ 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 M₂ 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 M₂ 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.