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.
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