  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 first illustrate our framework with ASPT phases
protected by fermion parity symmetry, then extend the classification of ASPT
phases to a general group extension. We demonstrate examples of explicit
construction of fixed-point LPDOs for ASPT phases including intrinsic ASPTs in
both $(1+1)D$ and $(2+1)D$ systems.
