  Thermodynamics is a highly successful macroscopic theory widely used across
the natural sciences and for the construction of everyday devices, from car
engines and fridges to power plants and solar cells. With thermodynamics
predating quantum theory, research now aims to uncover the thermodynamic laws
that govern finite size systems which may in addition host quantum effects.
Here we identify information processing tasks, the so-called "projections",
that can only be formulated within the framework of quantum mechanics. We show
that the physical realisation of such projections can come with a non-trivial
thermodynamic work only for quantum states with coherences. This contrasts with
information erasure, first investigated by Landauer, for which a thermodynamic
work cost applies for classical and quantum erasure alike. Implications are
far-reaching, adding a thermodynamic dimension to measurements performed in
quantum thermodynamics experiments, and providing key input for the
construction of a future quantum thermodynamic framework. Repercussions are
discussed for quantum work fluctuation relations and thermodynamic single-shot
approaches.
