Two destructive effects of decoherence on Bell inequality violation

We consider a system of two spin-$\frac{1}{2}$ particles, initially in an entangled Bell state. If one of the particles is interacting with an environment (e.g., a collection of $N$ independent spins), the two-particle system undergoes decoherence. Using a simple model of decoherence, we show that this process has two consequences. First, the maximal amount by which the Clauser-Horne-Shimony-Holt inequality is violated decays to zero. Second, the set of directions of measurement for which the inequality is violated is reduced in the course of decoherence. The volume of that set is bounded above by $\text{const}\times {\left|r\right|}^2$, where $r$ is the decoherence factor. We obtain similar results for the case when each of the two particles is in interaction with a separate environment.