  We discuss quantum state tomography via a stepwise reconstruction of the
eigenstates of the mixed states produced in experiments. Our method is tailored
to the experimentally relevant class of nearly pure states or simple mixed
states, which exhibit dominant eigenstates and thus lend themselves to low-rank
approximations. The developed scheme is applicable to any pure-state tomography
method, promoting it to mixed-state tomography. Here, we demonstrate it with
machine learning-inspired pure-state tomography based on neural-network
representations of quantum states. The latter have been shown to efficiently
approximate generic classes of complex (pure) states of large quantum systems.
We test our method by applying it to experimental data from trapped ion
experiments with four to eight qubits.
