Abstract
We apply the variational method to obtain the universal and analytical lower bounds for parameter precision in some noisy systems. We first derive a lower bound for phase precision in lossy optical interferometry at non-zero temperature. Then we consider the effect of both amplitude damping and phase diffusion on phase-shift precision. At last, we extend the constant phase estimation to the case of continuous fluctuating phase estimation, and find that due to photon losses the corresponding mean square error transits from the stochastic Heisenberg limit to the stochastic standard quantum limit as the total photon flux increases.
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