Tripartite Entanglement versus Tripartite Nonlocality in Three-Qubit Greenberger-Horne-Zeilinger-Class States

We analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three-qubit pure states in the Greenberger-Horne-Zeilinger class. We consider a family of states known as the generalized Greenberger-Horne-Zeilinger states and derive an analytical expression relating the three-tangle, which quantifies tripartite entanglement, to the Svetlichny inequality, which is a Bell-type inequality that is violated only when all three qubits are nonlocally correlated. We show that states with three-tangle less than $1/2$ do not violate the Svetlichny inequality. On the other hand, a set of states known as the maximal slice states does violate the Svetlichny inequality, and exactly analogous to the two-qubit case, the amount of violation is directly related to the degree of tripartite entanglement. We discuss further interesting properties of the generalized Greenberger-Horne-Zeilinger and maximal slice states.

Figure 1

Dots show a plot of Eq. (3) for $S_{\max }({\psi }_g)$ versus $\tau$ for the GGHZ states. Comparison to the numerical bounds [20] of Eq. (16) (solid line) shows agreement with the lower bound. Stars show a plot of Eq. (4) for $S_{\max }({\psi }_s)$ versus $\tau$ for the MS states. Comparison to the numerical bounds [20] of Eq. (16) (solid line) shows agreement with the upper bound.