Optimal quantum state tomography with known parameters

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 SICPOVMs, 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.