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arXiv:2512.09034 · Advanced Quantum Technologies

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

Gerard Anglès Munné, Paweł Cieśliński, Jan Wójcik, Wiesław Laskowski

quant-phSubmitted Dec 9, 2025• Announced Dec 11, 2025• 0 citations• Revised May 12, 2026 (v2)
v1 · Dec 9, 2025v2 · May 12, 2026
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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.

Journal reference
Advanced Quantum Technologies 9, no. 3 (2026): e01020
DOI
10.1002/qute.202501020
Comments
8 pages, 1 figure, few comments added

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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 NNN-partite system. Here, we generalize this result to arbitrary (N−k)(N-k)(N−k)-partite subsystems. We demonstrate that a single NNN-qubit state can violate all (Nk)\binom{N}{k}(kN​) relevant Bell inequalities simultaneously. We further construct an NNN-qubit Bell inequality, obtained by symmetrizing the (N−k)(N-k)(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.
Entanglement and Bell nonlocality have different sharing constraints across subsystems of a multipartite quantum system. We examine simultaneous violations of Bell inequalities with two-outcome observables and two or three measurement settings per party. We identify configurations whose simultaneous violations can be demonstrated on IBM Quantum hardware, and derive bounds and trade-off relations for several polygamous configurations. The results connect experimental capabilities of quantum computers with mathematical methods for studying the limits of quantum correlations.
The entangled multipartite systems, specially in pure states, exhibit the phenomenon of entanglement monogamy. Such systems also display the phenomenon of Bell nonlocality. Like entanglement monogamy relations, there are Bell monogamy relations. These relations suggest a sharing of nonlocality across the subsystems. The nonlocality, as characterized by Bell inequalities, of one subsystem limits the nonlocality exhibited by another subsystem. We show that the Bell monogamy relations can be violated by using local filtering operations. We consider permutation-symmetric multipartite pure states, in particular WWW states, to demonstrate the violation.
Quantum mechanics imposes limits on the statistics of certain observables. Perhaps the most famous example is the uncertainty principle. Similar trade-offs also exist for the simultaneous violation of multiple Bell inequalities. In the simplest case of three observers, it has been shown that if two observers violate a Bell inequality, then none of them can violate any Bell inequality with the third observer, a property called monogamy of Bell violations. Forms of Bell monogamy have been linked to the no-signaling principle, and the inability of simultaneous violations of all inequalities is regarded as their fundamental property. Here, we show that the Bell monogamy does not hold universally and that in fact the only monogamous situation exists for only three observers. Consequently, the nature of quantum nonlocality is truly polygamous. We present a systematic methodology for identifying quantum states, measurements, and tight Bell inequalities that do not obey the monogamy principle for any number of more than three observers. The identified polygamous inequalities enable any subset of [Formula: see text] observers to reveal nonlocality, which is also shown experimentally by measuring Bell-type correlations of six-photon Dicke states. Our findings may be exploited for multiparty quantum key distribution as well as simultaneous self-testing of multiple nodes in quantum networks.
Quantum nonlocality is a significant feature in quantum information theory, prompting recent investigations into the potential reuse of postmeasurement states to uncover nonlocality among sequentially measuring observers. While prior studies primarily focused on bipartite or tripartite systems and observers with one chain, such as multiple Bobs with a single Alice or multiple Charlies with a single Alice and Bob, our work extends beyond this framework. We explore sequential nonlocality in systems comprising more parties and observer chains. Our findings reveal that in nnn-partite systems, regardless of whether it is a single-chain or double-chain scenario, there exist unbounded sequential observers capable of detecting nonlocality through violations of the Mermin inequality. In contrast to the conjecture that sequential Bell nonlocality cannot manifest with multiple Alices and Bobs in bipartite systems [i.e., the double-chain setting, see S. M. Cheng et al., Phys. Rev. A 104, L060201 (2021)], our results suggest that increasing the number of subsystems may enable more observer chains to detect nonlocality alongside single observers. Our study advances research on sequential nonlocality, providing valuable insights into its detection across diverse scenarios.
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Quantum nonlocality is a significant feature in quantum information theory, prompting recent investigations into the potential reuse of post-measurement states to uncover nonlocality among sequentially measuring observers. While prior studies primarily focused on bipartite or tripartite systems and observers with one chain, such as multiple Bobs with a single Alice or multiple Charlies with a single Alice and Bob, our work extends beyond this framework. We explore sequential nonlocality in systems comprising more parties and observer chains. Our findings reveal that in nnn-partite systems, regardless of whether it is a single-chain or double-chain scenario, there exist unbounded sequential observers capable of detecting nonlocality through violations of the Mermin inequality. In contrast to the conjecture that sequential Bell nonlocality cannot manifest with multiple Alices and Bobs in bipartite systems (i.e., the double-chain setting)[Phys. Rev. A 104, L060201 (2021)], our results suggest that increasing the number of subsystems may enable more observer chains to detect nonlocality alongside single observers. Our study advances research on sequential nonlocality, providing valuable insights into its detection across diverse scenarios.