Optimal conditions for Bell-inequality violation in the presence of decoherence and errors

We obtain the set of all detector configurations providing the maximal violation of the Bell inequality in the Clauser–Horne–Shimony–Holt form for a general (pure or mixed) state of two qubits. Next, we analyze optimal conditions for the Bell-inequality violations in the presence of local decoherence, which includes energy relaxation at the zero temperature and arbitrary pure dephasing. We reveal that in most cases the Bell inequality violation is maximal for the “even” two-qubit state. Combined effects of measurement errors and decoherence on the Bell inequality violation are also discussed.