  Measuring the fluctuations of work in coherent quantum systems is notoriously
problematic. Aiming to reveal the ultimate source of these problems, we demand
of work measurement schemes the sheer minimum and see if those demands can be
met at all. We require ($\mathfrak{A}$) energy conservation for arbitrary
initial states of the system and ($\mathfrak{B}$) the Jarzynski equality for
thermal initial states. By energy conservation we mean that the average work
must be equal to the difference of initial and final average energies, and that
untouched systems must exchange deterministically zero work. Requirement
$\mathfrak{B}$ encapsulates the second law of thermodynamics and the
quantum--classical correspondence principle. We prove that work measurement
schemes that do not depend on the system's initial state satisfy $\mathfrak{B}$
if and only if they coincide with the famous two-point measurement scheme,
thereby establishing that state-independent schemes cannot simultaneously
satisfy $\mathfrak{A}$ and $\mathfrak{B}$. Expanding to the realm of
state-dependent schemes allows for more compatibility between $\mathfrak{A}$
and $\mathfrak{B}$. However, merely requiring the state-dependence to be
continuous still effectively excludes the coexistence of $\mathfrak{A}$ and
$\mathfrak{B}$, leaving the theoretical possibility open for only a narrow
class of exotic schemes.
