  Entanglement can be considered as a special quantum correlation, but not the
only kind. Even for a separable quantum system, it is allowed to exist
non-classical correlations. Here we propose two dissipative schemes for
generating a maximally correlated state of two qubits in the absence of quantum
entanglement, which was raised by [F. Galve, G. L. Giorgi, and R. Zambrini,
{\color{blue}Phys. Rev. A {\bf 83}, 012102 (2011)}]. These protocols take full
advantages of the interaction between four-level atoms and strongly lossy
optical cavities. In the first scenario, we alternatively change the phases of
two classical driving fields, while the second proposal introduces a strongly
lossy coupled-cavity system. Both schemes can realize all Lindblad terms
required by the dissipative dynamics, guaranteeing the maximally quantum
dissonant state to be the unique steady state for a certain subspace of system.
Moreover, since the target state is a mixed state, the performance of our
method is evaluated by the definition of super-fidelity $G(\rho_{1},\rho_{2})$,
and the strictly numerical simulations indicate that fidelity outstripping
$99\%$ of the quantum dissonant state is achievable with the current cavity
quantum electrodynamics parameters.
