  In classical thermodynamic processes the unavoidable presence of
irreversibility, quantified by the entropy production, carries two energetic
footprints: the reduction of extractable work from the optimal, reversible
case, and the generation of a surplus of heat that is irreversibly dissipated
to the environment. Recently it has been shown that in the quantum regime an
additional quantum irreversibility occurs that is linked to decoherence into
the energy basis. Here we employ quantum trajectories to construct
distributions for classical heat and quantum heat exchanges, and show that the
heat footprint of quantum irreversibility differs markedly from the classical
case. We also quantify how quantum irreversibility reduces the amount of work
that can be extracted from a state with coherences. Our results show that
decoherence leads to both entropic and energetic footprints which both play an
important role in the optimization of controlled quantum operations at low
temperature.
