Determine the minimum achievable figures of merit for statistical light analysis

Determine the minimal achievable values of the cumulative-sum and cumulative-product figures of merit for architecture-agnostic coherence-matrix eigenvalue extraction, as these minima depend on the statistical nature of the incident light.

Background

The paper introduces two intensity-only optimization objectives for diagonalizing an incident optical coherence matrix: a cumulative-sum figure of merit, L(1)\mathcal{L}^{(1)}, and a cumulative-product figure of merit, L(2)\mathcal{L}^{(2)}. Their optimization is intended to reveal the coherence-matrix eigenvalues using a programmable photonic circuit treated as a black box.

The authors note that zero is not generally the attainable optimum. In particular, the cumulative-sum objective reaches zero only for a fully coherent input, whereas partially coherent or incoherent inputs produce nonzero lower bounds. The paper leaves the exact minimum unspecified because it depends on the statistical properties of the input radiation.

References

Still, the optimal solution is not found when the FoMs reach zero, and the minimal value is unknown as it depends on the nature of the statistical light fed at the input. Indeed, \mathcal{L}{(1)}(\Phi) converges to zero only if the input is a fully coherent source.

— Black-Box Coherence Matrix Eigen-Spectroscopy with Programmable Photonics  (2608.19353 - Zelaya et al., 19 Aug 2026) in Results, subsection “Statistical Light Analysis via Photonics”