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Quantum detector tomography is a fundamental technique for calibratingquantum devices and performing quantum engineering tasks. In this paper, wedesign optimal probe states for detector estimation based on the minimum upperbound of the mean squared error (UMSE) and the maximum robustness. We establishthe minimum UMSE and the minimum condition number for quantum detectors andprovide concrete examples that can achieve optimal detector tomography. Inorder to enhance the estimation precision, we also propose a two-step adaptivedetector tomography algorithm to optimize the probe states adaptively based ona modified fidelity index. We present a sufficient condition on when theestimation error of our two-step strategy scales inversely proportional to thenumber of state copies. Moreover, the superposition of coherent states is usedas probe states for quantum detector tomography and the estimation error isanalyzed. Numerical results demonstrate the effectiveness of both the proposedoptimal and adaptive quantum detector tomography methods.

Optimal and two-step adaptive quantum detector tomography