Work statistics and quantum trajectories: No-click limit and non-Hermitian Hamiltonians

We investigate quantum work statistics within the standard two-point measurement scheme in continuously monitored quantum systems, including the effects of generalized unitary evolution, possibly controlled by quantum circuit models, and multiple generalized measurements as well as postselection of trajectories. We derive an explicit expression for the work generating function that naturally incorporates non-Hermitian dynamics arising from quantum jump processes and reveals deviations from the standard Jarzynski equality due to measurement-induced asymmetries. We illustrate our theoretical framework by analyzing a one-dimensional transverse-field Ising model under local spin monitoring. In this model, increased measurement strength projects the system onto the no-click state, leading to a suppression of energy fluctuations and measurement-induced energy saturation, reminiscent of the quantum Zeno effect. Moreover, we find signatures of the measurement-induced transition observed in the no-click limit in the moments of the work distribution.