A quantum state filter (QSF) is proposed in this paper to estimate a low-rank quantum density matrix from informationally incomplete and contaminated measurements. There exist sparse disturbances on the quantum density matrix and Gaussian noise in the measurements. A proximal Jacobian variant of the alternating direction method of multipliers (PJ-ADMM) is proposed to design the QSF. The closed-form solutions to three resulting subproblems are given and the iterative QSF is developed. The proposed QSF is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state-of-the-art methods.