Entropy of the quantum work distribution

We study properties of the work distribution of a many-body system driven through a quantum phase transition for both sudden quenches and finite time ramps. We show how the information entropy of these distributions provides a strong signature of the phase transitions, characteristics which are not clearly evident in cumulants such as the mean and variance. We provide general bound on this entropy in terms of both the thermodynamic entropy and the relative entropy of coherence and demonstrate the applicability of our results for models containing both quantum and localisation phase transitions. Further interesting characteristics are shown to be examinable through the non-Gaussianity of the distribution, which we characterize through two quantitative metrics: the skewness and the negentropy. For the quantum Ising model we she that a finite duration of the ramp enhances the non-Gaussianity of the distribution for a finite size system.