  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.
