The ongoing developments of analog and digital quantum simulators spark the interest in defining and classifying mixed-state phases of matter out of equilibrium. This consideration motivates the recent interest in symmetry-protected topological mixed states. In this work, we explore the physics of open system dynamics whose non-equilibrium steady states are the decohered cluster states —- symmetry-protected topological mixed states protected by the strong Z2 and weak Z2 symmetries.

We examine the applicability of the usual diagnostics for the symmetry-protected topological pure states to the non-equilibrium steady mixed states, including the projective representation, string order parameters, edge modes, etc. The open system dynamics can be understood as a defect diffusion-annihilation process, giving us a physical understanding of the mixing time. We examine the stability/instability of the non-equilibrium steady states under perturbations. Finally, we propose a potential realization of such dynamics on a quantum simulator.