  This investigation continues a program aiming at obtaining effective quantum
models to describe measurement statistics. In [Stark, arXiv:1209.5737 (2012)],
we have described how the Gram matrix associated to the prepared states and the
performed POVM elements can be estimated via convex relaxation of a rank
minimization problem. This Gram matrix determines the density matrices and the
POVM elements uniquely up to simultaneous rotations (with respect to the
Hilbert-Schmidt inner product) in the vector space of Hermitian matrices.
However, when the description of the experiment needs to be connected to
textbook quantum mechanics, explicit expressions for the states and the POVM
elements in terms of positive semidefinite matrices are required. In this
paper, we describe a heuristic algorithm that takes the state-meaurement Gram
matrix as input and searches explicit realizations of this Gram matrix in terms
of quantum mechanically valid density matrices and POVM elements.
