Title: Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States

Abstract: We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological (SPT) phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in $(1+1)D$ and $(2+1)D$ systems, unveiling two distinct types of injectivity conditions inherent in short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. We illustrate our framework with examples of ASPTs protected by fermion parity symmetry in both $(1+1)D$ and $(2+1)D$ systems. In addition, we discuss the classification of ASPT phases for a general group extension. We demonstrate examples of explicit construction of fixed-point LPDOs for ASPT phases including intrinsic ASPTs in $(1+1)D$ systems.