  One of the major goals of quantum thermodynamics is the characterization of
irreversibility and its consequences in quantum processes. Here, we discuss how
entropy production provides a quantification of the irreversibility in open
quantum systems through the quantum fluctuation theorem. We start by
introducing a two-time quantum measurement scheme, in which the dynamical
evolution between the measurements is described by a completely positive,
trace-preserving (CPTP) quantum map (forward process). By inverting the
measurement scheme and applying the time-reversed version of the quantum map,
we can study how this backward process differs from the forward one. When the
CPTP map is unital, we show that the stochastic quantum entropy production is a
function only of the probabilities to get the initial measurement outcomes in
correspondence of the forward and backward processes. For bipartite open
quantum systems we also prove that the mean value of the stochastic quantum
entropy production is sub-additive with respect to the bipartition (except for
product states). Hence, we find a method to detect correlations between the
subsystems. Our main result is the proposal of an efficient protocol to
determine and reconstruct the characteristic functions of the stochastic
entropy production for each subsystem. This procedure enables to reconstruct
even others thermodynamical quantities, such as the work distribution of the
composite system and the corresponding internal energy. Efficiency and possible
extensions of the protocol are also discussed. Finally, we show how our
findings might be experimentally tested by exploiting the state-of-the-art
trapped-ion platforms.
