# Trade-offs in multi-party Bell inequality violations in qubit networks

## TL;DR

This paper explores the trade-offs and monogamy relations in Bell inequality violations within qubit networks, revealing how multiple inequalities can be simultaneously violated and identifying conditions for genuine multi-party non-locality.

## Contribution

It systematically analyzes monogamy relations in multipartite Bell inequalities, deriving tight trade-off bounds and extending understanding beyond uncertainty relations.

## Key findings

- Weak monogamy relations allow simultaneous violations of multiple Bell inequalities.
- A tight trade-off relation was derived that cannot be explained solely by uncertainty principles.
- Genuine multi-party non-locality exhibits monogamy, unlike some bipartite violations.

## Abstract

Two overlapping bipartite binary input 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.