We discuss the uniqueness of quantum states compatible with the mean values of a set of observables. For a given pure state, two different types of uniqueness are considered: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known previously that, for a d-dimensional Hilbert space, there exists a set of 4d-5 observables that uniquely determines any pure state. We show that for case (2), 5d-7 observables suffice to uniquely determine any pure state. Equivalences and differences between case (1) and (2) are also discussed under certain conditions. Most of our discussions will generalize easily to mixed states of low rank.