Extracting filtered signal statistics of continuously measured quantum systems

The joint state of a continuously monitored quantum system and the classical filtered measurement record has recently been shown to be described by a quantum Fokker-Planck master equation [Phys. Rev. Lett. 129, 050401 (2022)]. We present a deterministic approach to compute the steady state of the system and detector. The method is shown to become particularly efficient in the absence of feedback, which we exploit to develop a perturbative approach valid for weak feedback. We show that through this method we can extract the full counting statistics of the signal, the quantum-classical mutual information between the system and signal, and the Fisher information of the signal, which can be used for sensing applications. Our results are illustrated with both single-qubit models and the spin chains governed by the one-dimensional transverse field Ising model or the Lipkin-Meshkov-Glick model.