  We consider a quasi-probability distribution of work for an isolated quantum
system coupled to the energy-storage device given by the ideal weight.
Specifically, we analyze a trade-off between changes in average energy and
changes in weight's variance, where work is extracted from the coherent and
incoherent ergotropy of the system. Primarily, we reveal that the extraction of
positive coherent ergotropy can be accompanied by the reduction of work
fluctuations (quantified by a variance loss) by utilizing the non-classical
states of a work reservoir. On the other hand, we derive a
fluctuation-decoherence relation for a quantum weight, defining a lower bound
of its energy dispersion via a dumping function of the coherent contribution to
the system's ergotropy. Specifically, it reveals that unlocking ergotropy from
coherences results in high fluctuations, which diverge when the total coherent
energy is unlocked. The proposed autonomous protocol of work extraction shows a
significant difference between extracting coherent and incoherent ergotropy:
The former can decrease the variance, but its absolute value diverges if more
and more energy is extracted, whereas for the latter, the gain is always
non-negative, but a total (incoherent) ergotropy can be extracted with finite
work fluctuations. Furthermore, we present the framework in terms of the
introduced quasi-probability distribution, which has a physical interpretation
of its cumulants, is free from the invasive nature of measurements, and reduces
to the two-point measurement scheme (TPM) for incoherent states. Finally, we
analytically solve the work-variance trade-off for a qubit, explicitly
revealing all the above quantum and classical regimes.
