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Topological order offers possibilities for processing quantum informationwhich can be immune to imperfections. However, the question of its stabilityout of equilibrium is relevant for experiments, where coupling to anenvironment is unavoidable. In this work we demonstrate the robustness ofcertain aspects of $Z_2 \times Z_2$ symmetry-protected topological (SPT) orderagainst a wide class of dissipation channels in the Lindblad and quantumtrajectory formalisms of an open quantum system. This is illustrated using theone-dimensional $ZXZ$ cluster Hamiltonian along with Pauli-string jumpoperators. We show that certain choices of dissipation retaining strongsymmetries support a steady-state manifold consisting of two non-local logicalqubits, and for Hamiltonian perturbations preserving the global symmetry, themanifold remains long-lived. In contrast, this metastability is destroyed uponbreaking the above-mentioned symmetry. While the localized edge qubits of thecluster Hamiltonian are not conserved by the Lindbladian evolution, they docorrespond to weak symmetries and thus retain a memory of their initial stateat all times in the quantum trajectories. We utilize this feature to constructprotocols to retrieve the quantum information either by monitoring jumps orerror mitigation. Our work thus proposes a novel framework to study thedynamics of dissipative SPT phases and opens the possibility of engineeringentangled states relevant to quantum information processing.

Edge modes and symmetry-protected topological states in open quantum systems