  The standard definition of quantum fluctuating work is based on the
two-projective energy measurement, which however does not apply to systems with
initial quantum coherence because the first projective energy measurement
destroys the initial coherence, and affects the subsequent evolution of the
system. To address this issue, several alternative definitions, such as those
based on the full counting statistics and the Margenau-Hill distribution, have
been proposed recently. These definitions seem ad hoc because justifications
for them are still lacking. In the current study, by utilizing the quantum
Feynman-Kac formula and the phase space formulation of quantum mechanics, we
prove that the leading order of work distributions is equal to the classical
work distribution. Thus we prove the validity of the quantum-classical
correspondence of work distributions for initial states with quantum coherence,
and provide some justification for those definitions of work. We use an exactly
solvable model of the linearly dragged harmonic oscillator to demonstrate our
main results.
