De las Cuevas, G.Cubitt, J.L.Wolf, M.M.Pérez García, David2023-06-182023-06-1820160022248810.1063/1.4954983https://hdl.handle.net/20.500.14352/24628We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.engjournal articlehttp://scitation.aip.org/content/aip/journal/jmp/57/7/10.1063/1.4954983restricted access517Análisis matemático1202 Análisis y Análisis Funcional