  Quantum coherence marks a deviation from classical physics, and has been
studied as a resource for metrology and quantum computation. Finding reliable
and effective methods for assessing its presence is then highly desirable.
Coherence witnesses rely on measuring observables whose outcomes can guarantee
that a state is not diagonal in a known reference basis. Here we experimentally
measure a novel type of coherence witness that uses pairwise state comparisons
to identify superpositions in a basis-independent way. Our experiment uses a
single interferometric set-up to simultaneously measure the three pairwise
overlaps among three single-photon states via Hong-Ou-Mandel tests. Besides
coherence witnesses, we show the measurements also serve as a Hilbert-space
dimension witness. Our results attest to the effectiveness of pooling many
two-state comparison tests to ascertain various relational properties of a set
of quantum states.
