In classical thermodynamic processes the unavoidable presence of irreversibility, quantified by the entropy production, carries two energetic footprints: the reduction of extractable work from the optimal, reversible case, and the generation of a surplus of heat that is irreversibly dissipated to the environment. Recently it has been shown that in the quantum regime an additional quantum irreversibility occurs that is linked to decoherence into the energy basis. Here we employ quantum trajectories to construct distributions for classical heat and quantum heat exchanges, and show that the heat footprint of quantum irreversibility differs markedly from the classical case. We also quantify how quantum irreversibility reduces the amount of work that can be extracted from a state with coherences. Our results show that decoherence leads to both entropic and energetic footprints which both play an important role in the optimization of controlled quantum operations at low temperature.

在经典的热力学过程中，不可避免的不可逆性的存在（通过熵的产生来量化）带有两个能量足迹：从最佳的可逆情况中减少可提取功，以及产生不可逆地散发到环境中的多余热量。最近已经显示出，在量子状态中，发生了另外的量子不可逆性，该不可逆性与退相干成能量基础有关。在这里，我们使用量子轨迹来构建经典热交换和量子热交换的分布，并表明量子不可逆性的热足迹与经典情况明显不同。我们还量化了量子不可逆性如何减少可以从具有相干状态的状态中提取的功。