Fundamental limitations in the purifications of tensor networks

TL;DR

This paper reveals a fundamental limitation in representing certain quantum mixed states with tensor networks, showing some states cannot be purified in a translationally invariant form across all system sizes.

Contribution

It proves that some mixed states represented by TI MPDOs lack a corresponding TI purification, highlighting intrinsic constraints in tensor network descriptions of quantum states.

Findings

Existence of mixed states without TI purification

Limitations apply to classical states as well

Proof based on undecidable problems and canonical forms

Abstract

We 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 (MPDO) 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.