Nonlocality and entanglement for symmetric states

In this paper, building on some recent progress combined with numerical techniques, we shed some light on how the nonlocality of symmetric states is related to their entanglement properties and on their potential usefulness in quantum information processing. We use semidefinite programming techniques to devise a device-independent classification of three 4-qubit states into two classes inequivalent under local unitaries and permutation of systems. We study nonlocal properties when the number of parties grows large for two important classes of symmetric states: $W$ states and GHZ states, showing that they behave differently under the inequalities we consider. We also discuss the monogamy arising from the nonlocal correlations of symmetric states. We show that although monogamy in a strict sense is not guaranteed for all symmetric states, strict monogamy is achievable for all Dicke states when the number of parties goes to infinity.

Figure 1

A comparison of $\frac{1}{2^{Eg}}$ (filled circles), the violation of ${\mathcal{P}}^4$ (filled stars), and ${\mathcal{Q}}_3^4$ (open squares) for states $|000\theta \rangle$, when $\theta$ varies from 0 to $\pi$.

Figure 2

(a) The tetrahedron state and (b) the state $|000+\rangle$ in the Majorana representation.

Figure 3

(a) The tetrahedron state and (b) the 4-qubit GHZ state in the Majorana representation.

Figure 4

Violations of ${\mathcal{P}}^n$ by the state $|{\mathrm{GHZ}}_n\rangle$ (squares), with the numerical violations of ${\mathcal{P}}^n$ (upward-pointing triangles) and ${\mathcal{Q}}_{n-1}^n$ (downward-pointing triangles) of $|W_n\rangle$ as a function of $n$ (number of parties), compared to $\frac{1}{2^{Eg\left(\right|W_n\rangle )}}$ (circles) and $\frac{1}{2^{Eg\left(\right|{\mathrm{GHZ}}_n\rangle )}}$(diamonds).

Figure 5

A comparison of the violation of $\mathcal{L}$ (29) for states $\left|S\right(n,\frac{n}{2})\rangle$ (squares) and $|W_n\rangle$ (circles) as a function of $n$ (the number of parties).