Unveiling average symmetry-protected topological phases through tensor network states

Tensor network states serve as a natural framework for characterizing the topological phases of matter, and handling the decoherence of the quantum states in open quantum systems. In this talk, we will introduce the tensor network representation of average symmetry-protected topological (ASPT) phases by locally-purified density operators. In this framework, we can unambiguously define the exact and average symmetries and produce the complete classification of ASPT phases in (1+1)D and (2+1)D systems without choosing a specific basis.