How pure can we go with adiabatic state manipulation?

Dissipative systems with decoherence free subspaces, a.k.a. dark spaces (DSs), can be used to protect quantum information. At the same time, dissipation is expected to give rise to coherent information degradation outside the DS. Employed to support quantum information platforms, DSs can be adiabatically modified in a way that resembles adiabatic control of coherent systems. Here we study the slow evolution of a purely dissipative system with a spectral gap $\gamma$, characterized by a strong symmetry, under a cyclic protocol with period $T$. Non-adiabatic corrections to the state evolution give rise to decoherence: the evolution within the instantaneous DS is described by a time-local effective Liouvillian operator that leads to purity degradation over a period, of order $1/\gamma T$. We obtain a closed form of the latter to order $1/(\gamma T)^2$. Our analysis underlines speed limitations in quantum information processing in the absence of corrective measures.