Comparative analysis of convex optimization and maximum likelihood estimation for shot-noise-affected positive operator-valued measure estimation in photon-number-resolving detectors

Photon-number-resolving detectors play a crucial role in quantum optics and quantum information processing. Their characterization is typically expressed through a positive operator-valued measure (POVM), making accurate estimation methods that reliably link the POVM to measurement outcomes essential. However, shorter measurement times result in fewer detection events, rendering photon counting measurements highly susceptible to shot noise. Consequently, statistical fluctuations limit estimation accuracy. In this study, we numerically investigate the impact of limited data on POVM estimation through simulations. We compare two estimation approaches: least-squares-based convex optimization and maximum likelihood estimation. By incorporating appropriately chosen constraints, convex optimization mitigates shot-noise-induced instabilities. In contrast, maximum likelihood estimation, although limited, achieves superior performance without relying on such additional constraints. To validate these findings, we conduct experiments that use a superconducting transition-edge sensor as a photon-number-resolving detector. The experimental results show strong agreement with numerical predictions, confirming the reliability of our simulation framework.