  We study the role of the initial quantum coherence in coherent processes
generated by an external control of some parameters by looking on the
thermodynamic work done. We start by taking in exam an active state and we
isolate the contribution to the ergotropy coming from the quantum coherence
among the energy eigenstates. It is shown to be related to the quantum relative
entropy of coherence through an inequality which involves the completely
passive state connected to the initial state. Finally, we extend the analysis
to a general out-of-equilibrium process and we show how to take in account in
the statistics of work the effects of the initial quantum coherence.
