  Quantum detector tomography is a fundamental technique for calibrating
quantum devices and performing quantum engineering tasks. In this paper, we
design optimal probe states for detector estimation based on the minimum upper
bound of the mean squared error (UMSE) and the maximum robustness. We establish
the minimum UMSE and the minimum condition number for quantum detectors and
provide concrete examples that can achieve optimal detector tomography. In
order to enhance the estimation precision, we also propose a two-step adaptive
detector tomography algorithm to optimize the probe states adaptively based on
a modified fidelity index. We present a sufficient condition on when the
estimation error of our two-step strategy scales inversely proportional to the
number of state copies. Moreover, the superposition of coherent states is used
as probe states for quantum detector tomography and the estimation error is
analyzed. Numerical results demonstrate the effectiveness of both the proposed
optimal and adaptive quantum detector tomography methods.
