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The simultaneous multi-parameter estimation problem using a class ofmulti-mode entangled states is investigated in this paper. Specifically, theproblem of optical phase imaging is considered and the quantum probe is takento be a balanced coherent superposition of components with an arbitrary quantumstate in one mode and vacuum states in all the other modes, which is ageneralization of the multi-mode NOON state. The analytical form for thequantum Cramer-Rao bound (QCRB) is presented, which shows the performance byproviding a lower bound of the estimation uncertainty. It is shown that theNOON state has the worst performance among those in the class of the entangledstates considered. We also analyze in detail four different scenarios, whichare the NOON state, the entangled coherent state, the entangled squeezedcoherent state, and the entangled squeezed vacuum state. From the comparisonamong these four states, we find that when the mean photon number is fixed, thesqueezed vacuum state has the smallest QCRB, followed by the squeezed coherentstate, entangled coherent state, and NOON state. We also illustrate that thebalanced entangled state can perform better than a more generalized unbalancedform studied in previous works for certain scenarios. Finally, we give anexperimental setup for producing a two-mode entangled state that can beat theNOON state in quantum metrology.

Quantum multi-parameter estimation with generalized balanced multi-mode NOON-like states