  Recent studies have pointed out the intrinsic dependence of figures of merit
of thermodynamic relevance -- such as work, heat and entropy production -- on
the amount of quantum coherences that is made available to a system. However,
whether coherences hinder or enhance the value taken by such quantifiers of
thermodynamic performance is yet to be ascertained. We show that, when
considering entropy production generated in a process taking a finite-size
bipartite quantum system out of equilibrium through local non-unitary channels,
no general monotonicity relationship exists between the entropy production and
degree of quantum coherence in the state of the system. A direct correspondence
between such quantities can be retrieved when considering specific forms of
open-system dynamics applied to suitably chosen initial states. Our results
call for a systematic study of the role of genuine quantum features in the
non-equilibrium thermodynamics of quantum processes.
