entropy flows between the system and its environment. To separate the two contributions, we usually write the rate of change of the entropy $S$ of a system as $$\label{dSdt}
\frac{{\,\mathrm{d}}S}{{\,\mathrm{d}}t} = \Pi - \Phi,$$ where $\Phi$ is the entropy flux rate from the system to the environment and $\Pi$ is the entropy production rate. According to the second law of thermodynamics, we should have $\Pi \geq 0$ and $\Pi = 0$ if and only if the system is in equilibrium. Hence, the entropy production rate may be used as a natural quantifier of the degree of irreversibility of a process. For this reason, a thorough understanding of the entropy production rate is both fundamentally relevant and technologically desirable. On the one hand, such understanding would provide the much needed foundation for the emergence of time-symmetry breaking entailed by irreversibility and epitomized, for instance, by seminal results such as Onsager’s theory of irreversible currents [@Onsager1931; @*Onsager1931a; @Machlup1953; @DeGroot1961; @Tisza1957]. On the other hand, a characterization of irreversible entropy could help us to design thermodynamically efficient quantum technologies [@Escher2011; @Verstraete2008a].

The description of entropy production in open quantum systems is still an open question, despite substantial progress [@Spohn1978; @Breuer2003; @Deffner2011; @Santos2017b; @Brunelli2016a; 