Xiao, ShuixinLiang, WeichaoWang, YuanlongDong, DaoyiPetersen, Ian R.Ugrinovskii, Valery2026-06-112026-06-110018-9286https://hdl.handle.net/1885/733810288The estimation of all the parameters in an unknown quantum state or measurement device, commonly known as quantum state tomography (QST) and quantum detector tomography (QDT), is crucial for comprehensively characterizing and controlling quantum systems. In this paper, we introduce a framework, in two different bases, that utilizes multiple quantum processes to simultaneously identify a quantum state and a detector. We develop a closed-form algorithm for this purpose and prove that the mean squared error (MSE) scales as O(1/N) for both QST and QDT, where N denotes the total number of state copies. This scaling aligns with established patterns observed in previous works that addressed QST and QDT as independent tasks. Furthermore, we formulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, where the physical constraints of the state and detector are characterized by polynomial equalities and inequalities. The effectiveness of our proposed methods is validated through numerical examples.This research was supported by the Australian Research Council (DP200102945, DP210101938, FT220100656, DP240101494), the Innovation Program for Quantum Science and Technology 2023ZD0301400, and the National Natural Science Foundation of China (12288201). Shuixin Xiao would like to gratefully acknowledge the support from the IEEE Control Systems Society Graduate Collaboration Fellowship. The material in this paper was partially presented at the 63rd IEEE Conference on Decision and Control, Milan, Italy, December 16-19, 2024. (Corresponding authors: Yuanlong Wang, Daoyi Dong).enPublisher Copyright: © 1963-2012 IEEE.Quantum detector tomographyquantum state tomographyquantum system identificationsum of squares202510.1109/TAC.2025.3635368105022698332