The Richness of Bell Nonlocality: Generalized Bell Polygamy and Hyper-Polygamy

TL;DR

This paper explores the extensive and scalable nature of Bell nonlocality in multipartite quantum states, revealing generalized polygamy and hyper-polygamy phenomena that enhance understanding of quantum correlations.

Contribution

It generalizes Bell polygamy to arbitrary subsystems, introduces a symmetrized Bell inequality maximally violated by such states, and demonstrates advantages over GHZ states in multipartite nonlocality.

Findings

A single N-qubit state can violate all N choose kell inequalities simultaneously.

Symmetrized Bell inequalities are maximally violated by states exhibiting generalized polygamy.

Polygamy and hyper-polygamy provide advantages for scalable quantum nonlocality certification.

Abstract

Non-classical quantum correlations underpin both the foundations of quantum mechanics and modern quantum technologies. Among them, Bell nonlocality is a central example. For bipartite Bell inequalities, nonlocal correlations obey strict monogamy: a violation of one inequality precludes violations of other inequalities on the overlapping subsystems. In the multipartite setting, however, Bell nonlocality becomes inherently polygamous. This was previously shown for subsystems obtained by removing a single particle from an $N$-partite system. Here, we generalize this result to arbitrary $(N-k)$-partite subsystems. We demonstrate that a single $N$-qubit state can violate all $\binom{N}{k}$ relevant Bell inequalities simultaneously. We further construct an $N$-qubit Bell inequality, obtained by symmetrizing the $(N-k)$-qubit ones, that is maximally violated by states exhibiting this generalized polygamy. We compare these violations with those achievable by GHZ states and show that polygamy offers an advantage in multipartite scenarios, providing new insights into scalable certification of non-classicality in quantum devices. Our analysis relies on symmetry properties of the MABK inequalities. Finally, we show that this behavior can occur across multiple subsystem sizes, a phenomenon we call hyper-polygamy. These structures reveal the remarkable abundance of nonlocality present in multipartite quantum states and offer perspectives for their applications in quantum technologies.