Bell inequalities from group actions: Three parties and non-Abelian groups

In a previous publication, we showed how group actions can be used to generate Bell inequalities. The group action yields a set of measurement probabilities whose sum is the basic element in the inequality. The sum has an upper bound if the probabilities are a result of a local, realistic theory, but this bound can be violated if the probabilities come from quantum mechanics. In our previous paper, we considered the case of only two parties making the measurements and single-generator groups. Here we show that the method can be extended to three parties, and it can also be extended to non-Abelian groups. We discuss the resulting inequalities in terms of nonlocal games.

Figure 1

The three different bases. Solid is $|\pm x\rangle$, dashed is $\left\{\right|u_0\rangle ,|u_1\rangle \}$, and dotted is $\left\{\right|v_0\rangle ,|v_1\rangle \}$.