  As a generalization of the standard phase retrieval problem, we seek to
reconstruct symmetric rank-1 matrices from inner products with subclasses of
positive semidefinite matrices. For such subclasses, we introduce random
cubatures for spaces of multivariate polynomials based on moment conditions.
The inner products with samples from sufficiently strong random cubatures allow
the reconstruction of symmetric rank-1 matrices with a decent probability by
solving the feasibility problem of a semidefinite program.
