  The entropy produced when a quantum system is driven away from equilibrium
can be decomposed in two parts, one related with populations and the other with
quantum coherences. The latter is usually based on the so-called relative
entropy of coherence, a widely used quantifier in quantum resource theories. In
this paper we argue that, despite satisfying fluctuation theorems and having a
clear resource-theoretic interpretation, this splitting has shortcomings.
First, it predicts that at low temperatures the entropy production will always
be dominated by the classical term, irrespective of the quantum nature of the
process. Second, for infinitesimal quenches, the radius of convergence diverges
exponentially as the temperature decreases, rendering the functions
non-analytic. Motivated by this, we provide here a complementary approach,
where the entropy production is split in a way such that the contributions from
populations and coherences are written in terms of a thermal state of a
specially dephased Hamiltonian. The physical interpretation of our proposal is
discussed in detail. We also contrast the two approaches by studying work
protocols in a transverse field Ising chain, and a macrospin of varying
dimension.
