Extended Fluctuation Theorems for Repeated Measurements and Feedback within Hamiltonian Framework

We derive the extended fluctuation theorems in presence of multiple measurements and feedback, when the system is governed by Hamiltonian dynamics. We use only the forward phase space trajectories in the derivation. However, to obtain an expression for the efficacy parameter, we must necessarily use the notion of reverse trajectory. Our results show that the correction term appearing in the exponent of the extended fluctuation theorems are non-unique, whereas the physical meaning of the efficacy parameter is unique.