Work Estimation and Work Fluctuations in the Presence of Non-Ideal Measurements*

From the perspective of quantum thermodynamics, realizable measurements cost work, and result in measurement devices that are not perfectly correlated with the measured systems. We investigate the consequences for the estimation of work in non-equilibrium processes and the fundamental structure of the fluctuations when one assumes that the measurements are non-ideal. We show that obtaining work estimates and their statistical moments at finite work cost implies an imperfection of the estimates themselves: more accurate estimates incur higher costs. We revisit these concepts and investigate the consequences for these quantities when one does not assume ideal measurements. We explicitly show how the average work is modified and discuss the operational meaning of the corresponding estimates. We show that while Jarzynski’s equality can be maintained exactly at the expense of a correction that only depends on the system’s Hamiltonian, the more general relation due to Crooks (as well as related results linking irreversibility and dissipation) no longer hold in the presence of non-ideal measurements. Our results provide a quantification of the cost of obtaining information about work as well as the trustworthiness of it.