  Matrix Product States form the basis of powerful simulation methods for
ground state problems in one dimension. Their power stems from the fact that
they faithfully approximate states with a low amount of entanglement, the "area
law". In this work, we establish the mixed state analogue of this result: We
show that one-dimensional mixed states with a low amount of entanglement,
quantified by the entanglement of purification, can be efficiently approximated
by Matrix Product Density Operators (MPDOs). In combination with results
establishing area laws for thermal states, this helps to put the use of MPDOs
in the simulation of thermal states on a formal footing.
