  Recent contributions in the field of quantum state tomography have shown
that, despite the exponential growth of Hilbert space with the number of
subsystems, tomography of one-dimensional quantum systems may still be
performed efficiently by tailored reconstruction schemes. Here, we discuss a
scalable method to reconstruct mixed states that are well approximated by
matrix product operators. The reconstruction scheme only requires local
information about the state, giving rise to a reconstruction technique that is
scalable in the system size. It is based on a constructive proof that generic
matrix product operators are fully determined by their local reductions. We
discuss applications of this scheme for simulated data and experimental data
obtained in an ion trap experiment.
