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.

　　Publication: 

　　Automatica, Volume 141, July 2022. 

　　Author: 

　　Shuixin Xiao 

　　University of Michigan – Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China 

　　School of Engineering and Information Technology, University of New South Wales, Canberra ACT 2600, Australia 

　　Email: xiaoshuixin@sjtu.edu.cn  

　　Yuanlong Wang 

　　Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia 

　　Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 

　　Email: wangyuanlong@amss.ac.cn  

　　Daoyi Dong 

　　School of Engineering and Information Technology, University of New South Wales, Canberra ACT 2600, Australia 

　　Email: daoyidong@gmail.com  

　　Jun Zhang 

　　University of Michigan – Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China 

　　Email: zhangjun12@sjtu.edu.cn