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An open question of fundamental importance in thermodynamics is how todescribe the fluctuations of work for quantum coherent processes. In thestandard approach, based on a projective energy measurement both at thebeginning and at the end of the process, the first measurement destroys anyinitial coherence in the energy basis. Here we seek for extensions of thisapproach which can possibly account for initially coherent states. We considerall measurement schemes to estimate work and require that (i) the difference ofaverage energy corresponds to average work for closed quantum systems, and that(ii) the work statistics agree with the standard two-measurement scheme forstates with no coherence in the energy basis. We first show that such a schemecannot exist. Next, we consider the possibility of performing collectivemeasurements on several copies of the state and prove that it is stillimpossible to satisfy simultaneously requirements (i) and (ii). Nevertheless,improvements do appear, and in particular we develop a measurement scheme whichacts simultaneously on two copies of the state and allows to describe a wholeclass of coherent transformations.

No-go theorem for the characterisation of work fluctuations in coherent quantum systems