Enhancing multi-step quantum state tomography by PhaseLift

Multi-photon system has been studied by many groups, however the biggest challenge faced is the number of copies of an unknown state are limited and far from detecting quantum entanglement. The difficulty to prepare copies of the state is even more serious for the quantum state tomography. One possible way to solve this problem is to use adaptive quantum state tomography, which means to get a preliminary density matrix in the first step and revise it in the second step. In order to improve the performance of adaptive quantum state tomography, we develop a new distribution scheme of samples and extend it to three steps, that is to correct it once again based on the density matrix obtained in the traditional adaptive quantum state tomography. Our numerical results show that the mean square error of the reconstructed density matrix by our new method is improved to the level from 10⁻⁴ to 10⁻⁹ for several tested states. In addition, PhaseLift is also applied to reduce the required storage space of measurement operator. - Highlights: • PhaseLift is firstly applied in the adaptive quantum state tomography and the required store space of measurement operator is reduced. • Traditional adaptive quantum state tomography is extended to three-step adaptive quantum state tomography. • A new scheme of copy distribution of state is proposed to effectively use the limited copies of state. • For several tested states, the mean square error of the reconstructed state is reduced to 10⁻⁹ from 10⁻⁴. • Fidelity is more accurately estimated with limited copies of state compared with traditional adaptive quantum state tomography.