  The entropy produced when a system undergoes an infinitesimal quench is
directly linked to the work parameter susceptibility, making it sensitive to
the existence of a quantum critical point. Its singular behavior at $T=0$,
however, disappears as the temperature is raised, hindering its use as a tool
for spotting quantum phase transitions. Notwithstanding the entropy production
can be split into classical and quantum components, related with changes in
populations and coherences. In this paper we show that these individual
contributions continue to exhibit signatures of the quantum phase transition,
even at arbitrarily high temperatures. This is a consequence of their intrinsic
connection to the derivatives of the energy eigenvalues and eigenbasis. We
illustrate our results in two prototypical quantum critical systems, the
Landau-Zener and $XY$ models.
