Initial quantum coherence in the thermodynamic work

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.