Fate of symmetry protected coherence in open quantum system

We investigate the fate of coherence in the dynamical evolution of a symmetry protected quantum system. Under the formalism of system-plus-bath for open quantum system, the anti-unitary symmetry exhibits significant difference from the unitary one in protecting initial coherence. Specifically, taking advantage of Lindblad master equation, we find that a pure state in the symmetry protected degenerate subspace will decohere even though both the system Hamiltonian and system-environment interaction respect the same anti-unitary symmetry. In contrast, the coherence will persist when the protecting symmetry is unitary. We provide an elaborate classification table to illustrate what kinds of symmetry combinations are able to preserve the coherence of initial state, which is confirmed by several concrete models in spin-$3/2$ system. Our results could help to explore the possible experimental realization of stable time-reversal symmetric topological states.