Optimization of Quantum Measurements for Robustness Against Dark Counts: The $D$-Trace Optimality Framework

Quantum communication, while promising unparalleled security, faces significant practical challenges due to imperfections in quantum devices, particularly in single-photon detectors (SPDs). One of the key issues is the impact of dark counts, which cause erroneous detections and limit the effective range of quantum communication systems, such as in quantum key distribution (QKD). In this paper, we introduce a novel framework for addressing dark count errors through a new optimality criterion, termed $D$-trace optimality. This criterion aims to optimize quantum measurements to minimize the impact of dark counts without relying on hardware replacements. We propose an optimization scheme that transforms general measurements into $D$-trace optimal measurements, ensuring robustness against dark count effects. Our results suggest that, under certain conditions, this approach can improve the performance of quantum communication systems, especially in scenarios where the prior distribution of states is known. Although challenges remain in obtaining accurate prior distributions and ensuring compatibility with other strategies. This work lays the foundation for future research on optimizing quantum measurements to mitigate the effects of noise, including dark counts, in practice.