  This paper systematically studied heat transfer through two transversely
coupled qubits in contact with two types of heat reservoirs. One is the
independent heat reservoir which essentially interacts with only a single
qubit, the other is the common heat reservoir which is allowed to
simultaneously interact with two qubits. Compared to independent heat
reservoirs, common reservoirs always suppress heat current in most cases.
However, the common environment could enhance heat current, if the dissipation
rate corresponding to the higher eigenfrequency is significantly higher than
that corresponding to the lower eigenfrequency. In particular, in the case of
resonant coupling of two qubits and the proper dissipations, the steady state
can be decomposed into a stationary dark state which doesn't evolve and
contributes zero heat current, and a residual steady state which corresponds to
the maximal heat current. This dark state enables us to control steady-state
heat current with an external control field and design a thermal modulator. In
addition, we find that inverse heat currents could be present in the
dissipative subchannels between the system and reservoirs, which interprets the
suppression roles of common heat reservoirs. We also calculate the concurrence
of assistance (COA) of the system and find that heat current and COA have the
same trend with temperature, which further indicates that entanglement can be
regarded as a resource to regulate heat transport.
