Exergy of passive states: Waste energy after ergotropy extraction

Work extraction protocol is always a significant issue in the context of quantum batteries, in which the notion of ergotropy is used to quantify a particular amount of energy that can be extracted through unitary processes. Given the total amount of energy stored in a quantum system, quantifying wasted energy after the ergotropy extraction is a question to be considered when undesired coupling with thermal reservoirs is taken into account. In this paper, we show that some amount of energy can be lost when we extract ergotropy from a quantum system and quantified by the exergy of passive states. Through a particular example, one shows that ergotropy extraction can be done by preserving the quantum correlations of a quantum system. Our study opens the perspective for new advances in open system quantum batteries able to explore exergy stored as quantum correlations.