Singular Asymptotics of SPADE in Quantum Source Discrimination
We study far-field discrimination between one and two incoherent point sources in the singular regime of weak and closely spaced emitters. Under ideal alignment, spatial-mode demultiplexing (SPADE) attains the quantum-optimal large-sample Stein exponent, but the finite-photon behavior near the one-source boundary and the effect of realistic imperfections remain less understood. Using singular learning theory, we analyze both the aligned and misaligned problems. In the aligned Gaussian case, for prior densities that are smooth and strictly positive in the physical $(\epsilon,s)$ coordinates near the singularity of the aligned model, direct imaging and SPADE share the same real log canonical threshold $\lambda=1/2$ but have different multiplicities, yielding distinct Bayes free-energy asymptotics. A fixed nonzero misalignment removes the exact support mismatch of ideal SPADE: locally, the fixed-offset binary-SPADE Kullback--Leibler function has the normal-crossing form $K\asymp\epsilon^2s^2$, giving $(\lambda,m)=(1/2,2)$ under the same prior class, as for direct imaging. The local separation scale nevertheless depends on the source-position convention. Moreover, full Hermite--Gaussian mode counting about an offset sorter axis has a reflection-induced KL-zero branch at $s^\ast=2\theta$, where the two-source alternative becomes indistinguishable from the null. Pointwise finite-$n$ binary-SPADE calculations exhibit the corresponding power collapse. These results identify measurement support, nuisance geometry, prior weighting, and identifiability as structural ingredients that must be tracked in finite-photon quantum discrimination.