Trade-offs in multiparty Bell-inequality violations in qubit networks

Abstract

Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signalling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signalling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different number of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multi-party non-locality, as evidenced by a generalised Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.

Randomness versus nonlocality in the Mermin-Bell experiment with three parties

“…The linearisations in section 3 can also be read as inequalities identifying parts of the boundary of the set of quantum correlations. The results reveal that part of the boundary of the quantum set is flat, a characteristic that has previously been remarked upon in [35,37]. It may be interesting to study how our results generalise to Bell experiments involving more parties.…”

“…For readers familiar with both, it may seem natural to ask whether the security of the HBB scheme can be proved device independently, i.e., without assuming that the participants' devices are necessarily measuring σ x and σ y . There have indeed been proposals to design a device-independent secret-sharing protocol based on the GHZ-paradox or other correlations arising from GHZ states [20,34,35]. However, we found that the HBB scheme is completely insecure from a deviceindependent point of view.…”

Geometry of the set of quantum correlations

“…A more natural family of such functions was proposed by Slofstra [60], but these require a large number of measurement settings on each side. On the other hand, a simple example was recently found in the tripartite p3 -2 -2q scenario by Ramanathan and Mironowicz [61]. In this section we give an example in the bipartite scenario p2 -3 -2q and two additional examples in the p3 -2 -2q scenario.…”

Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems

“…Generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. The genuine multipartite nonlocality, as evidenced by a generalized Svetlichny inequality, does exhibit monogamy property [25]. There is a complementarity relation between dichotomic observables leading to the monogamy of Bell inequality violations [26].…”