This paper introduces a tensor network approach called the locally purified density operator (LPDO) to study symmetry-protected topological (SPT) phases in mixed states. The LPDO framework extends the concept of injectivity from matrix product states to mixed states, identifying two types of injectivity conditions that prevent long-range order. The authors classify decohered average SPT (ASPT) phases, consistent with previous results from spectrum sequence techniques. They demonstrate an intuitive construction of ASPT states using the decorated domain wall picture, and derive classification data and obstruction functions for ASPT phases protected by strong and weak symmetries. The framework is applied to both (1+1)D and (2+1)D systems, with examples of intrinsic ASPT phases in these dimensions. The LPDO formalism provides a clear physical interpretation of symmetry and topology in mixed states, enabling the classification of ASPT phases in open quantum systems. The paper also discusses the physical realization of ASPT phases in decohered and disordered systems, and generalizes the LPDO structure to higher dimensions. The results highlight the complexity and richness of quantum phases in mixed states, revealing new insights into the interplay between symmetry and topology in open quantum systems.This paper introduces a tensor network approach called the locally purified density operator (LPDO) to study symmetry-protected topological (SPT) phases in mixed states. The LPDO framework extends the concept of injectivity from matrix product states to mixed states, identifying two types of injectivity conditions that prevent long-range order. The authors classify decohered average SPT (ASPT) phases, consistent with previous results from spectrum sequence techniques. They demonstrate an intuitive construction of ASPT states using the decorated domain wall picture, and derive classification data and obstruction functions for ASPT phases protected by strong and weak symmetries. The framework is applied to both (1+1)D and (2+1)D systems, with examples of intrinsic ASPT phases in these dimensions. The LPDO formalism provides a clear physical interpretation of symmetry and topology in mixed states, enabling the classification of ASPT phases in open quantum systems. The paper also discusses the physical realization of ASPT phases in decohered and disordered systems, and generalizes the LPDO structure to higher dimensions. The results highlight the complexity and richness of quantum phases in mixed states, revealing new insights into the interplay between symmetry and topology in open quantum systems.