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

TL;DR

This paper tackles the problem of how Bell nonlocality can be shared across many subsystems in multipartite quantum states, challenging the traditional monogamy picture. By leveraging MABK correlations, permutation-invariant Dicke states, and symmetrized MABK constructions, it proves that for any $k>0$ there exists an $N_k$ such that a single $N_k$-qubit state can violate all $\binom{N_k}{k}$ two-setting Bell inequalities on all $(N_k-k)$-party subsystems, a property called $k$-polygamy. The authors derive the maximal violations, show optimal states $|\phi_N^k\rangle$, and demonstrate that for $k>1$ these polygamous states outperform GHZ-based strategies in aggregate nonlocality across subsystems. They further extend the idea to $K$-hyper-polygamy, identifying minimal $N_K$ for multiple $k$ values via SDP and constructing states $|\Psi^K_N\rangle$ that realize simultaneous violations, highlighting the rich structure and potential applications in device-independent certification and scalable quantum networks.

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 with $k>0$. 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.

Figures (1)

• Figure 1: Comparison of the sum of squared violation factors of the $(N-k)$-qubit MABK inequalities for the $\ket{\text{GHZ}_{N-k}}$ (dashed lines) and the optimal $k$-polygamous states $|\phi_N^{k} \rangle$ (solid lines). Each color corresponds to a different value of $k=1,2,3$. Despite the fact that the GHZ states give rise to the maximal violation of a given MABK inequality, the amount of small violations over all subsystems makes the polygamous states achieve higher values of $\mathcal{S_{\psi}}$.

Theorems & Definitions (3)

• Theorem 1
• Proposition 1
• proof