  The work is a concept of fundamental importance in thermodynamics. An open
question is how to describe the work fluctuation for quantum coherent processes
in the presence of initial quantum coherence in the energy basis. With the aim
of giving a unified description, here we introduce and study a class of
quasiprobability distributions of work, which give an average work equal to the
average energy change of the system and reduce to the two-projective
measurement scheme for an initial incoherent state. Moreover, we characterize
the work with the help of fluctuation relations. In particular, by considering
the joint distribution of work and initial quantum coherence, we find a
fluctuation theorem involving quantum coherence, from which follows a second
law of thermodynamics in the case of initial thermal populations. Furthermore,
we propose a way to measure the characteristic function of work and we discuss
the negativity of the quasiprobability. The effects of coherence are also
investigated for a simple system of a qubit.
