“…More strikingly, the single-phase estimation with the QCRB in the presence of noisy environments, e.g., photon loss [17][18][19], phase diffusion [20,21], and thermal noise [22,23], can be tackled using the variational method [17,20] pro-posed by Escher, greatly promoting the practical applications of quantum metrology [24][25][26]. On the other hand, extending toward the multiple phase estimation with the QCRB has attracted considerable interest more recently, thereby resulting in the potential applications [27][28][29][30][31][32][33][34], such as quantum-enhanced sensor network [29][30][31][32] and optical imaging [33,34]. Moreover, in order to improve the precision of multiple-phase estimation, multimode NOON (or NOON-like) states [35][36][37][38][39], generalized entangled coherent states [40] and multimode Gaussian states [41] have been considered, even in the presence of noisy environment [42][43][44][45].…”