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.
