  At non-zero temperature classical systems exhibit statistical fluctuations of
thermodynamic quantities arising from the variation of the system's initial
conditions and its interaction with the environment. The fluctuating work, for
example, is characterised by the ensemble of system trajectories in phase space
and, by including the probabilities for various trajectories to occur, a work
distribution can be constructed. However, without phase space trajectories, the
task of constructing a work probability distribution in the quantum regime has
proven elusive. Here we use quantum trajectories in phase space and define
fluctuating work as power integrated along the trajectories, in complete
analogy to classical statistical physics. The resulting work probability
distribution is valid for any quantum evolution, including cases with
coherences in the energy basis. We demonstrate the quantum work probability
distribution and its properties with an exactly solvable example of a driven
quantum harmonic oscillator. An important feature of the work distribution is
its dependence on the initial statistical mixture of pure states, which is
reflected in higher moments of the work. The proposed approach introduces a
fundamentally different perspective on quantum thermodynamics, allowing full
thermodynamic characterisation of the dynamics of quantum systems, including
the measurement process.
