  The ability of quantum states to be in superposition is one of the key
features that sets them apart from the classical world. This `coherence' is
rigorously quantified by resource theories, which aim to understand how such
properties may be exploited in quantum technologies. There has been much
research on what the resource theory of coherence can reveal about quantum
metrology, almost all of which has been from the viewpoint of Fisher
information. We prove however that the relative entropy of coherence, and its
recent generalisation to POVMs, naturally quantify the performance of Bayesian
metrology. In particular, we show how a coherence measure can be applied to an
ensemble of states. We then prove that during parameter estimation, the
ensemble relative entropy of coherence is equal to the difference between the
information gained, and the optimal Holevo information. We call this relation
the CXI equality. The ensemble coherence lets us visualise how much information
is locked away in superposition inaccessible with a given measurement scheme,
and quantify the advantage that would be gained by using a joint measurement on
multiple states. Our results hold regardless of how the parameter is encoded in
the state, encompassing unitary, dissipative, and discrete settings. We
consider both projective measurements, and general POVMs. This work suggests
new directions for research in quantum resource theories, provides a novel
operational interpretation for the relative entropy of coherence and its POVM
generalisation, and introduces a new tool to study the role of quantum features
in metrology.
