We address the question of whether open quantum system dynamics which host mixed symmetry-protected topological (SPT) states as steady states continue to do so after introducing symmetric perturbations. In particular, we discuss the characteristics of the decohered cluster state --- a mixed SPT protected by a combined strong and weak symmetry --- and construct a parent Lindbladian which hosts it as a steady state. The parent Lindbladian can be mapped onto reaction-diffusion dynamics, which is exactly solvable, even in the presence of certain perturbations. This allows us to study the steady states of the parent Lindbladian in detail, including states which, to the best of our knowledge, have not been previously discussed. Through a combination of analytical and numerical methods, we find that the perturbed Lindbladians exhibit strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbation strength for typical symmetric perturbations, leading to instability of the mixed steady-state SPT order. However, for perturbations which only introduce defects of the weak symmetry, we find that the mixed steady-state SPT order survives. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.

Pizza and drinks will be served after the seminar in ATL 2117.