Nonlocality of Symmetric States

In this Letter we study the nonlocal properties of permutation symmetric states of $n$ qubits. We show that all these states are nonlocal, via an extended version of the Hardy paradox and associated inequalities. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different nonlocal features. Belonging to a given entanglement class will guarantee the violation of associated Bell inequalities which see the persistence of correlations to subsets of players, whereas there are states outside that class which do not violate.

Figure 1

The state (a) $|T⟩=\sqrt{1/3}|0000⟩+\sqrt{2/3}|S(4,3)⟩$ does not violate $Q_3^4$ while all states with degeneracy $d=3$ do, such as the state (b) $|D_3⟩=K\sum _{\mathrm{perm}}|000+⟩$.