Efficient description of many-body systems with Matrix Product Density Operators

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.