  The geometrical arrangement of a set of quantum states can be completely
characterized using relational information only. This information is encoded in
the pairwise state overlaps, as well as in Bargmann invariants of higher degree
written as traces of products of density matrices. We describe how to measure
Bargmann invariants using suitable generalizations of the SWAP test. This
allows for a complete and robust characterization of the projective-unitary
invariant properties of any set of pure or mixed states. As applications, we
describe basis-independent tests for linear independence, coherence, and
imaginarity. We also show that Bargmann invariants can be used to characterize
multi-photon indistinguishability.
