Quantum state and detector tomography with known rank

Quantum state tomography and quantum detector tomography are two main problems in quantum system identification. In this paper, we study state tomography and detector tomography with prior knowledge about the true rank of the unknown state/detector. In this case, we propose a sufficient condition for static (non-adaptive) tomography algorithms to achieve the optimal infidelity scaling O(1/N) on the number of state copies N. Then we present two examples of such static quantum state tomography and quantum detector tomography algorithms. Numerical examples demonstrate the effectiveness of our algorithms.