  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.

Quantum Physics