We discuss the uniqueness of quantum states compatible with given results for

measuring a set of observables. For a given pure state, we consider two

different types of uniqueness: (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 is known 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. Thus there is a gap between the

results for (1) and (2), and we give some examples to illustrate this. The case

of observables corresponding to reduced density matrices (RDMs) of a

multipartite system is also discussed, where we improve known bounds on local

dimensions for case (2) in which almost all pure states are uniquely determined

by their RDMs. We further discuss circumstances where (1) can imply (2). We use

convexity of the numerical range of operators to show that when only two

observables are measured, (1) always implies (2). More generally, if there is a

compact group of symmetries of the state space which has the span of the

observables measured as the set of fixed points, then (1) implies (2). We

analyze the possible dimensions for the span of such observables. Our results

extend naturally to the case of low rank quantum states.