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

Understand

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.