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

Akio Yoshizawa, Tetsuya Tsuruta, Daiji Fukuda

Published May 1, 2026• 0 citations

Abstract

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.

DOI: 10.1063/5.0323841

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The accurate characterization of quantum systems is essential for the advancement of quantum technologies. In particular, certifying convex functions of quantum states plays a central role in many applications. We present a certification method for experimentally prepared quantum states that accounts for both shot noise and measurement imperfections in the data-acquisition stage. Building upon previous work, our method extends confidence regions to accommodate imperfect control over measurements. The values of the functions can then be bounded using convex optimization techniques. We provide explicit prescriptions for quantifying the noise contribution from finite statistics and for estimating the effect of measurement imperfections. By jointly incorporating statistical and systematic errors, the method yields a robust certification framework for quantum experiments.

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