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

An open question of fundamental importance in thermodynamics is how to describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek for extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems, and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to satisfy simultaneously requirements (i) and (ii). Nevertheless, improvements do appear, and in particular we develop a measurement scheme which acts simultaneously on two copies of the state and allows to describe a whole class of coherent transformations.