Expanding bipartite Bell inequalities for maximum multi-partite randomness

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.

💡 Research Summary

This paper investigates the fundamental trade-off between nonlocality and randomness certification in multipartite quantum systems. Focusing on the family of Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities, the authors analyze how much device-independent (DI) global randomness can be certified from quantum behaviors achieving a given MABK violation value.

The key findings reveal a stark difference between scenarios with an even or odd number of parties, N. For odd N, the authors show that maximum global randomness (N bits) can be certified for any value of MABK violation, using a newly constructed family of Bell inequalities. For even N, the situation is more constrained. They demonstrate that maximum randomness cannot be achieved beyond a certain threshold MABK violation value, m*. Below this threshold, maximum N-bit randomness is possible, but above it, the certifiable randomness monotonically decreases as the MABK violation increases. The paper provides a lower bound on this trade-off curve and presents numerical evidence suggesting it is tight. Furthermore, they show that this threshold m* approaches the maximum quantum value of the MABK inequality as N increases, indicating that the trade-off becomes less severe for larger even numbers of parties.

To derive these results, the authors’ main technical contribution is the development and enhancement of a method they term “Expanding Bell Inequalities.” This technique allows the construction of tailored multipartite Bell inequalities for randomness certification by using a bipartite Bell expression (e.g., CHSH) as a “seed.” The seed is summed over different pairs of parties while the remaining parties perform a fixed measurement. The core of their security analysis is a “Decoupling Lemma,” which proves that maximal violation of such an expanded inequality implies a tensor product structure between the honest parties’ post-measurement system and the adversary’s (Eve’s) system. This decoupling is the crucial condition for certifying maximum DI randomness, as it ensures Eve gains no information about the outputs.

In summary, this work precisely characterizes the limitations of the MABK inequalities for randomness generation, introduces powerful new families of Bell inequalities that overcome these limitations, and provides a general method—expanding bipartite seeds with a decoupling guarantee—for translating insights from the well-understood bipartite case into the complex multipartite domain for DI cryptography.