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We present a general framework to study the simultaneous estimation ofmultiple phases in the presence of noise as a discretized model for phaseimaging. This approach can lead to nontrivial bounds of the precision formultiphase estimation. Our results show that simultaneous estimation (SE) ofmultiple phases is always better than individual estimation (IE) of each phaseeven in noisy environment. However with $d$ being the number of phases, the$O(d)$ advantage in the variance of the estimation, with which SE outperformsIE schemes for noiseless processes, may disappear asymptotically. When noise islow, those bounds recover the Heisenberg scale with the $O(d)$ advantage. Theutility of the bound of multiple phase estimation for photon loss channels isexemplified.

Quantum-enhanced metrology for multiple phase estimation with noise