  We find necessary and sufficient conditions to determine the
inter-convertibility of quantum systems under time-translation covariant
evolution, and use it to solve several problems in quantum thermodynamics both
in the single-shot and asymptotic regimes. It is well known that the resource
theory of quantum athermality is not reversible, but in PRL 111, 250404 (2013)
it was claimed that the theory becomes reversible ``provided a sublinear amount
of coherent superposition over energy levels is available". Here we show that
if a sublinear amount of coherence among energy levels were considered free,
then the resource theory of athermality would become trivial. Instead, we show
that by considering a sublinear amount of energy to be free, the theory of
athermality becomes reversible for the pure-state case. A proof of the same
claim for the mixed-state case is still lacking.
