Abstract: Modern scientific studies often involve complex quantum systems, and scientists need to learn the systems from experimental data. As density matrices are usually employed to characterize the quantum states of the systems, this paper investigates estimation of density matrices. We propose statistical methodologies to construct density matrix estimators and establish an asymptotic theory for the estimation methods. We show that the proposed density matrix estimators are consistent and have good convergence rates. A numerical study is conducted to demonstrate the finite sample performances of the proposed estimators.

Key words and phrases: Convergence rate, density matrix, large matrix estimation, quantum system, quantum tomography, spectral norm.