  In quadratic fermionic models we determine a quantum correction to the work
statistics after a sudden and a time-dependent driving. Such a correction lies
in the non-commutativity of the initial quantum state and the time-dependent
Hamiltonian, and is revealed via the Kirkwood-Dirac quasiprobability (KDQ)
approach to two-times correlators. Thanks to the latter, one can assess the
onset of non-classical signatures in the KDQ distribution of work, in the form
of negative and complex values that no classical theory can reveal. By applying
these concepts on the one-dimensional transverse-field Ising model, we relate
non-classical behaviours of the KDQ statistics of work in correspondence of the
critical points of the model. Finally, we also prove the enhancement of the
extracted work in non-classical regimes where the non-commutativity takes a
role.
