SPT phases for mixed states in one dimension

This study explores the specific structure of 1D SPT phases with $G\times H$ symmetry ($G, H$ are finite non-Abelian groups) and constructs an order parameter. We demonstrate that this order parameter can comprehensively describe this specific structure. Utilizing this order parameter, we prove that when the SPT phase theory of symmetry $G$ is extended to mixed states, if the density matrix is treated as a pure SPT state with $G\times G$ symmetry, the Hermiticity and positive semi-definiteness ensure that the SPT phase of the mixed state is similarly described by $H^2(G)$. Finally, we propose that under this perspective, the SPT phase of time-reversal symmetry no longer exists in mixed states.