  Quantum coherence is an indispensable resource for quantum technological
applications. It is known to be distillable from a noisy form using operations
that cannot create coherence. However, distillation exacts a hidden coherent
measurement cost, whose extent has not previously been estimated. Here we show
that this cost (quantified by an equivalent number of Hadamard measurements) is
related to what we call the irretrievable coherence: the difference between the
coherence of formation and the distillable coherence. We conjecture (and make
partial progress towards proving) that when distilling from many copies of a
given noisy coherent state, the coherent measurement cost scales extensively in
the number of copies, at an asymptotic rate exactly equalling the input's
irretrievable coherence. This cost applies to any application whereof coherence
distillation is an incidental outcome (e.g. incoherent randomness extraction),
but the implications are more dramatic if pure coherence is the only desired
outcome: the measurement cost may often be higher than the distilled yield, in
which case coherence should rather be prepared afresh than distilled from a
noisy input.
