Classification and monogamy of three-qubit biseparable Bell correlations

We strengthen the set of Bell-type inequalities presented by Sun and Fei [Phys. Rev. A 74, 032335 (2006)] that give a classification for biseparable correlations and entanglement in tripartite quantum systems. We will furthermore consider the restriction to local orthogonal spin observables and show that this strengthens all previously known such tripartite inequalities. The quadratic inequalities we find indicate a type of monogamy of maximal biseparable tripartite quantum correlations, although the nonmaximal ones can be shared. This is contrasted to recently found monogamy inequalities for bipartite Bell correlations in tripartite systems.

Figure 1

${\mathcal{D}}_3^{\left(i\right)}-{\mathcal{D}}_3^{(i+1)}$ plane with the stronger bound given by the circle with radius $\sqrt{5∕2}$ which strengthens the less strong bound with radius $\sqrt{3}$ that is given by the dashed circle.

Figure 2

The results of Theorem 3 for orthogonal observables. For comparison to the case where the observables were not restricted to be orthogonal, the dashed square is included that has edge length $2\sqrt{2}$ and which is the largest square in Fig. 1.