Stochastic thermodynamics based on incomplete information: Generalized Jarzynski equality with measurement errors with or without feedback

In the derivation of fluctuation relations, and in stochastic thermodynamics in general, it is tacitly assumed that we can measure the system perfectly, i.e., without measurement errors. We here demonstrate for a driven system immersed in a single heat bath, for which the classic Jarzynski equality $\langle e^{-β(W-ΔF)}\rangle = 1$ holds, how to relax this assumption. Based on a general measurement model akin to Bayesian inference we derive a general expression for the fluctuation relation of the measured work and we study the case of an overdamped Brownian particle and of a two-level system in particular. We then generalize our results further and incorporate feedback in our description. We show and argue that, if measurement errors are fully taken into account by the agent who controls and observes the system, the standard Jarzynski-Sagawa-Ueda relation should be formulated differently. We again explicitly demonstrate this for an overdamped Brownian particle and a two-level system where the fluctuation relation of the measured work differs significantly from the efficacy parameter introduced by Sagawa and Ueda. Instead, the generalized fluctuation relation under feedback control, $\langle e^{-β(W-ΔF)-I}\rangle = 1$, holds only for a superobserver having perfect access to both the system and detector degrees of freedom, independently of whether or not the detector yields a noisy measurement record and whether or not we perform feedback.