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We consider a quasi-probability distribution of work for an isolated quantumsystem coupled to the energy-storage device given by the ideal weight.Specifically, we analyze a trade-off between changes in average energy andchanges in weight's variance, where work is extracted from the coherent andincoherent ergotropy of the system. Primarily, we reveal that the extraction ofpositive coherent ergotropy can be accompanied by the reduction of workfluctuations (quantified by a variance loss) by utilizing the non-classicalstates of a work reservoir. On the other hand, we derive afluctuation-decoherence relation for a quantum weight, defining a lower boundof its energy dispersion via a dumping function of the coherent contribution tothe system's ergotropy. Specifically, it reveals that unlocking ergotropy fromcoherences results in high fluctuations, which diverge when the total coherentenergy is unlocked. The proposed autonomous protocol of work extraction shows asignificant difference between extracting coherent and incoherent ergotropy:The former can decrease the variance, but its absolute value diverges if moreand more energy is extracted, whereas for the latter, the gain is alwaysnon-negative, but a total (incoherent) ergotropy can be extracted with finitework fluctuations. Furthermore, we present the framework in terms of theintroduced quasi-probability distribution, which has a physical interpretationof its cumulants, is free from the invasive nature of measurements, and reducesto the two-point measurement scheme (TPM) for incoherent states. Finally, weanalytically solve the work-variance trade-off for a qubit, explicitlyrevealing all the above quantum and classical regimes.

Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction