Improving coherence with nested environments

We have in mind a register of qubits for an quantum information system, and consider its decoherence in an idealized but typical situation. Spontaneous decay and other couplings to the far environment considered as the world outside the quantum apparatus will be neglected, while couplings to quantum states within the apparatus, i.e. to a near environment are assumed to dominate. Thus the central system couples to the near environment which in turn couples to a far environment. Considering that the dynamics in the near environment is not sufficiently well known or controllable, we shall use random matrix methods to obtain analytic results. We consider a simplified situation where the central system suffers weak dephasing from the near environment, which in turn is coupled randomly to the far environment. We find the anti-intuitive result that increasing the coupling between near and far environment actually protects the central qubit.