量子源鉴别中SPADE的奇异渐近性
Singular Asymptotics of SPADE in Quantum Source Discrimination
AI总结：
研究弱且间距小的发射源在奇异区域中，使用空间模式解复用（SPADE）进行远场鉴别的有限光子行为，通过奇异学习理论分析对齐和未对齐情况下的贝叶斯自由能渐近性和Neyman-Pearson检验性能。
摘要：
我们研究在弱且间距小的发射源的奇异区域中，一个和两个非相干点源之间的远场鉴别。在理想对齐下，空间模式解复用（SPADE）达到了量子最优的大样本Stein指数，但靠近单源边界的有限光子行为以及现实缺陷的影响仍不太清楚。使用奇异学习理论，我们分析了对齐和未对齐问题。在对齐高斯情况下，我们推导了直接成像和SPADE的zeta函数极点，表明两者共享相同的实对数规范阈值$λ=1/2$但多重性不同，并得到了相应的贝叶斯自由能渐近性。这揭示了在对齐局部先验加权区域中，SPADE具有普遍的子领先优势。在未对齐设置中，我们研究了一个物理动机的二进制SPADE简化，它保留了靠近对齐时的完整领先$O(s^2)$泄漏对比，而详细的高阶模式重新分布的修正仅在$O(s^4)$阶出现。我们表明，未对齐的二进制SPADE和直接成像在不同的内在尺度上获得非平凡的局部幂，分别为$s=O(n^{-1/4})$和$s=O(n^{-1/2})$。然而，在常见物理条件下的有限$n$ Neyman-Pearson比较表明，直接成像在绘制的网格上更强，且未对齐的二进制SPADE表现出一个精确的盲分离$s^\ast=2θ$，其幂降至$α$。这些结果将模型奇异性识别为有限光子量子鉴别的结构组织原则，并阐明了理想对齐的SPADE基准如何在未对齐情况下无法转化为有限$n$优势。
英文摘要：
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 $(ε,s)$ coordinates near the singularity of the aligned model, direct imaging and SPADE share the same real log canonical threshold $λ=1/2$ 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 $K\asympε^2s^2$, giving $(λ,m)=(1/2,2)$ 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 $s^\ast=2θ$, 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.
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