Squeezing Enhanced Two Phase Estimation With N Particle W States

Abstract

Simultaneous estimation of two optical phases in a three‐mode interferometer assisted by optical parametric amplification (OPA) is investigated. The normally ordered characteristic function formalism is employed to derive analytical expressions for all photon number moments of the output state, allowing explicit evaluation of the quantum Fisher information matrix for multiparameter estimation. In the lossless regime, uniformly applied OPA significantly improves the attainable precision compared with an unamplified interferometer. As the OPA gain increases, the quantum Cramér–Rao bound gradually approaches Heisenberg scaling, resulting from the combined effects of increased photon resources and squeezing‐induced modification of photon number statistics. Examination of second‐order correlation functions indicates that this enhancement originates from amplified intra mode photon number correlations, rather than inter mode correlations. The analysis is extended to lossy interferometers using a purification‐based variational approach. Although photon loss reduces the achievable precision, the OPA assisted scheme preserves a clear advantage in the moderate loss regime, demonstrating robustness against dissipation. These findings clarify the physical origin of OPA enhanced performance and provide guidance for optimizing multiparameter phase estimation protocols in realistic noisy settings.