The adiabatic theorem and shortcuts to adiabaticity for the adiabatic dynamics of time-dependent decoherence-free subspaces are explored in this paper. Starting from the definition of the dynamical stable decoherence-free subspaces, we show that, under a compact adiabatic condition, the quantum state follows time-dependent decoherence-free subspaces (the adiabatic decoherence free subspaces) into the target subspace with extremely high purity, even though the dynamics of the quantum system may be non-adiabatic. The adiabatic condition mentioned in the adiabatic theorem is very similar with the adiabatic condition for closed quantum systems, except that the operators required to be slowness is on the Lindblad operators. We also show that the adiabatic decoherence-free subspaces program depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems has to be engineered according to the incoherent control program. Besides, the shortcuts to adiabaticity for the adiabatic decoherence-free subspaces program is also presented based on the transitionless quantum driving method. Finally, we provide an example of physical systems that support our programs. Our approach employs Markovian master equations and applies primarily to finite-dimensional quantum systems.
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