  Quantum work fluctuation theorems are known to hold when the work is defined as the difference between the outcomes of projective measurements carried out on the Hamiltonian of the system at the initial and the final time instants of the experimental realization of the process. A recent study showed that the theorem breaks down if the measurement is of a more general nature, i.e. if a positive operator valued measurement is used, and the deviation vanishes only in the limit where the operators become projective in nature. We study a simple two-state system subjected to a unitary evolution under a Hamiltonian that is linearly dependent on time, and verify the validity of the above statement. We further define a weak value of work and show that the deviation from the exact work fluctuation theorems are much less in this formalism. 
Exploring the extent of validity of quantum work fluctuation theorems in   the presence of weak measurements