Singular Asymptotics of SPADE in Quantum Source Discrimination

TL;DR

This paper analyzes the finite-photon behavior and imperfections in quantum source discrimination using SPADE, revealing how model singularities influence performance and the impact of misalignment.

Contribution

It provides a detailed singular learning theory analysis of SPADE's performance near the one-source boundary, including effects of misalignment and finite photon numbers.

Findings

Aligned SPADE attains the quantum-optimal Stein exponent asymptotically.

Misaligned binary-SPADE exhibits a blind separation at s* = 2θ.

Direct imaging outperforms misaligned binary-SPADE in finite-n Neyman-Pearson tests.

Abstract

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, we derive the zeta-function poles for direct imaging and SPADE, show that both share the same real log canonical threshold $\lambda=1/2$ but differ in multiplicity, and obtain the corresponding Bayes free-energy asymptotics. This yields a universal subleading advantage of aligned SPADE in the local prior-weighted regime. In the misaligned setting, we study a physically motivated binary-SPADE reduction that retains the full leading $O(s^2)$ leakage contrast near alignment, with corrections from the detailed higher-mode redistribution entering only at $O(s^4)$. We show that misaligned binary-SPADE and direct imaging acquire nontrivial local power on different intrinsic scales, $s=O(n^{-1/4})$ and $s=O(n^{-1/2})$, respectively. However, finite-$n$ Neyman--Pearson comparisons under common physical conditions reveal that direct imaging is stronger on the plotted grids and that misaligned binary-SPADE exhibits an exact blind separation $s^\ast=2\theta$, where its power collapses to $\alpha$. These results identify model singularity as a structural organizing principle for finite-photon quantum discrimination and clarify how ideal aligned SPADE benchmarks can fail to translate into finite-$n$ advantages under misalignment.