  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.
