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Characterizing fluctuations of work in coherent quantum systems isnotoriously problematic. We reveal the ultimate source of the problem byproving that ($\mathfrak{A}$) energy conservation and ($\mathfrak{B}$) theJarzynski fluctuation theorem cannot be observed at the same time. Condition$\mathfrak{A}$ stipulates that, for any initial state of the system, themeasured average work must be equal to the difference of initial and finalaverage energies, and that untouched systems must exchange deterministicallyzero work. Condition $\mathfrak{B}$ is only for thermal initial states andencapsulates the second law of thermodynamics. We prove that $\mathfrak{A}$ and$\mathfrak{B}$ are incompatible for work measurement schemes that aredifferentiable functions of the state and satisfy two mild structuralconstraints. This covers all existing schemes and leaves the theoreticalpossibility of jointly observing $\mathfrak{A}$ and $\mathfrak{B}$ open onlyfor a narrow class of exotic schemes. For the special but important case ofstate-independent schemes, the situation is much more rigid: we prove that,essentially, only the two-point measurement scheme is compatible with$\mathfrak{B}$.

Energy conservation and fluctuation theorem are incompatible for quantum work