  In this work, we study the stochastic entropy production in open quantum
systems whose time evolution is described by a class of non-unital quantum
maps. In particular, as in [Phys. Rev. E 92, 032129 (2015)], we consider Kraus
operators that can be related to a nonequilibrium potential. This class
accounts for both thermalization and equilibration to a non-thermal state.
Unlike unital quantum maps, non-unitality is responsible for an unbalance of
the forward and backward dynamics of the open quantum system under scrutiny.
Here, concentrating on observables that commute with the invariant state of the
evolution, we show how the non-equilibrium potential enters the statistics of
the stochastic entropy production. In particular, we prove a fluctuation
relation for the latter and we find a convenient way of expressing its average
solely in terms of relative entropies. Then, the theoretical results are
applied to the thermalization of a qubit with non-Markovian transient, and the
phenomenon of irreversibility mitigation, introduced in [Phys. Rev. Research 2,
033250 (2020)], is analyzed in this context.
