Expanding bipartite Bell inequalities for maximum multi-partite randomness

TL;DR

This paper investigates the relationship between Bell inequality violations and the maximum certifiable randomness in multi-partite quantum systems, introducing a new technique called 'expanding Bell inequalities' to optimize randomness certification.

Contribution

It introduces 'expanding Bell inequalities', a novel method to transform bipartite Bell expressions into multi-partite inequalities for enhanced randomness certification.

Findings

Maximum randomness is limited beyond a certain violation threshold for even-party systems.

Maximum randomness can be achieved for any violation in odd-party systems.

The new technique enables tailored Bell inequalities for multi-party randomness certification.

Abstract

Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities". Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.