Fig. 1. An example of two point sources with partial coherence. Two point objects are illuminated by an incoherent optical source. Even though the illumination source is completely incoherent, photons arriving at two points in the object plane may share the common origin, which exhibits partial coherence.

Fig. 2. QFI for the estimation of the separation of two partially coherent sources with different degrees of coherence p; the FI of SPADE equals the QFI.

Fig. 3. QFI and FI of direct imaging with the intensity measurement F_d for the estimation of the separation of two partially coherent sources with separation-dependent degree of coherence p; even though QFI drops to zero as the separation approaches zero, in the sub-Rayleigh region, where s < 2, QFI is much larger than F_d. QFI and FI meet at large separation.

Fig. 4. FI for SPADE with finite mode number k; even b-SPADE with k = 0 has an FI larger than F_d in the sub-Rayleigh region. b-SPADE performs worse than direct imaging in a large separation region and they are equal to each other at s₁ ≈ 2.47.

Fig. 5. Protocol of the adaptive measurement method. An initial estimation of the separation s is first obtained by direct imaging. By comparing the estimate s_est and s₁, we can choose the b-SPADE or direct imaging. At each step, the choice of different measurement methods is based on the estimation result calculated by the posterior distribution drawn from the last step. N is the number of cycles.

Fig. 6. Simulation results for adaptive measurement method conditioned on N = 1000 detected photons; dashed lines are the FI of direct imaging and b-SPADE, and stars are the simulation results calculated by the inverse of the mean squared error.