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.
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