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We discuss the problem of estimating a frequency via N-qubit probesundergoing independent dephasing channels that can be continuously monitoredvia homodyne or photo-detection. We derive the corresponding analyticalsolutions for the conditional states, for generic initial states and forarbitrary efficiency of the continuous monitoring. For the detection strategiesconsidered, we show that: i) in the case of perfect continuous detection, thequantum Fisher information (QFI) of the conditional states is equal to the oneobtained in the noiseless dynamics; ii) for smaller detection efficiencies, theQFI of the conditional state is equal to the QFI of a state undergoing the(unconditional) dephasing dynamics, but with an effectively reduced noiseparameter.

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