Quantum frequency estimation with conditional states of continuously monitored independent dephasing channels

We discuss the problem of estimating a frequency via N-qubit probes undergoing independent dephasing channels that can be continuously monitored via homodyne or photo-detection. We derive the corresponding analytical solutions for the conditional states, for generic initial states and for arbitrary efficiency of the continuous monitoring. For the detection strategies considered, we show that: (i) in the case of perfect continuous detection, the quantum Fisher information (QFI) of the conditional states is equal to the one obtained in the noiseless dynamics; (ii) for smaller detection efficiencies, the QFI of the conditional state is equal to the QFI of a state undergoing the (unconditional) dephasing dynamics, but with an effectively reduced noise parameter