Entanglement versus Bell nonlocality of quantum nonequilibrium steady states

We study the entanglement and the Bell nonlocality of a coupled two-qubit system, in which each qubit is coupled with one individual environment. We study how the nonequilibrium environments (with different temperatures or chemical potentials) influence the entanglement and the Bell nonlocality. Dependent on the inter-qubit coupling strength (relatively weak or strong compared to local qubits' frequencies) or the environmental nature (bosonic or fermionic), the two-qubit steady state can have strong correlations and violate the Bell inequalities with two or three measurements per party. Equilibrium environments compared to the nonequilibrium environments (with fixed mean temperatures or chemical potentials) do not give the maximal entanglement or the maximal violation of Bell inequalities if the two qubits are not identical, such as the two qubits having an energy detuning or coupling to the environment with unbalanced weights. The nonequilibrium conditions (characterized by the temperature differences) which give the maximal violation of Bell inequalities are different from the nonequilibrium conditions which give the maximal entanglement. The entanglement and the Bell nonlocality have different responses to the nonequilibrium environments. The spatial asymmetric two-qubit system coupled with nonequilibrium bosonic environments shows the thermal rectification effect, which can be witnessed by the Bell nonlocality. Our study demonstrates that the nonequilibrium environments are both valuable for the entanglement and Bell nonlocality resources, based on different optimal nonequilibrium conditions though.