Quantum-enhanced metrology for multiple phase estimation with noise

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