The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples. Article QUANTUM INFORMATION PROCESSING. 14(10):3983-3999 (2015)