  It is a well-known fact that the optimal POVM for quantum state tomography is
the symmetric, informationally complete, positive operator valued measure
(SIC-POVM). We investigate the same problem only in the case when there are
some a priori information about the state, specifically when some parameters
are known. In this paper we mainly focus on solving a 3-dimensional
optimization problem, which gives us a non-trivial example for the so-called
conditional SIC- POVMs, a straightforward generalization of the concept of
SIC-POVMs. We also present other special cases to show further applications of
the proposed numerical methods and to illustrate the complexity of this topic.
