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Autonomous entanglement engines have recently been proposed to generatesteady-state bipartite and multipartite entanglement exploiting only incoherentinteractions with thermal baths at different temperatures. In this work, weinvestigate the interplay between heat current and entanglement in a two-qubitentanglement engine, deriving a critical heat current for successful operationof the engine, i.e. a cut-off above which entanglement is present. The heatcurrent can thus be seen as a witness to the presence of entanglement. In theregime of weak-inter qubit coupling, we also investigate the effect of twoexperimentally relevant parameters for the qubits, the energy detuning andtunnelling, on the entanglement production. Finally, we show that the regime ofstrong inter-qubit coupling provides no clear advantage over the weak regime,in the context of out-of-equilibrium entanglement engines.

Critical heat current for operating an entanglement engine