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arXiv:2601.08924

Exploring Bell Nonlocality with Extremal Non-Signaling Boxes

Emmanuel Zambrini Cruzeiro, Junior R. Gonzales-Ureta, Raman Choudhary, Hugo Abreu, Adán Cabello, Sébastien Designolle

quant-phSubmitted Jan 13, 2026• Announced Jan 15, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Jan 13, 2026v2 · Oct 5, 2026
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Abstract

Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.

Comments
10 pages, 2 tables, 2 figures

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Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.
Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.
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Given a pair of isolated devices that accept random binary inputs and return binary outputs, a user can deduce from the observed data alone if the underlying mechanism can be explained classically. Bell's theorem further states that a classical explanation can be ruled out if the devices perform certain measurements on an entangled quantum state, underpinning the security of cryptographic protocols that are device-independent (DI). For certain protocols, such as those performed in a tight space, it might be difficult to perfectly enforce the non-signaling assumption required in Bell's theorem. This prompts the question: is quantum nonlocality robust to such imperfections? We show that if a binary channel sends a noisy copy of one party's input to the other before any measurements take place, the answer is yes. We completely characterize the vertices and facets of the local polytope in this scenario, and identify Bell inequalities that certify non-signaling quantum correlations. This is possible even when a near perfect copy of the input is sent. We go on to show that the identified inequalities are more robust to depolarizing noise than the Clauser-Horne-Shimony-Holt inequality when certifying DI randomness in this scenario. In addition, we characterize the local polytope when both parties receive a noisy copy of each other's input and make similar conclusions, leaving many new potential Bell inequalities to be explored.
Abstract Fundamental investigations in non-locality have shown that while the no-signaling principle alone is not sufficient to single out the set of quantum non-local correlations, local quantum mechanics and no-signaling together exactly reproduce the set of quantum correlations in the two-party Bell scenario. Here, we introduce and study an intermediate hybrid no-signaling quantum set of non-local correlations that we term HNSQ in the multi-party Bell scenario where some subsystems are locally quantum while the remaining subsystems are only constrained by the no-signaling principle. Specifically, the set HNSQ is a super-quantum set of correlations derived from no-signaling assemblages by performing quantum measurements on the trusted subsystems. We show that in contrast to the set NS of no-signaling behaviors, there exist extreme points of HNSQ in the tripartite Bell scenario that admit quantum realization. As a tool for optimization over the set HNSQ, we introduce an outer hierarchy of semi-definite programming approximations to the set following an approach put forward by Doherty–Parrilo–Spedalieri. We perform an extensive numerical analysis of the maximal violation of the facet Bell inequalities in the three-party binary input–output scenario and study the corresponding self-testing properties. In contrast to the usual no-signaling correlations, the new set allows for simple security proofs of (one-sided)-device-independent applications against super-quantum adversaries.
Activation of Bell nonlocality refers to the phenomenon that some entangled mixed states that admit a local hidden variable model in the standard Bell scenario nevertheless reveal their nonlocal nature in more exotic measurement scenarios. We present such a scenario that involves broadcasting the local subsystems of a single-copy of a bipartite quantum state to multiple parties, and use the scenario to study the nonlocal properties of the two-qubit isotropic state: ρα=α|Φ+⟩⟨Φ+|+(1−α)14. We present two main results, considering that Nature allows for (i) the most general no-signalling correlations, and (ii) the most general quantum correlations at the level of any hidden variable theory. We show that the state does not admit a local hidden variable description for α>0.559 and α>12 , in cases (i) and (ii) respectively, which in both cases provides a device-independent certification of the entanglement of the state. These bounds are significantly lower than the previously best-known bound of 0.697 for both Bell nonlocality and device-independent entanglement certification using a single copy of the state. Our results show that strong examples of non-classicality are possible with a small number of resources.
FINDING: The Popescu-Rohrlich (PR) box defines the maximal nonlocal vertex of the Bell polytope; its extremal structure is characterized by convex geometry, and recent work shows PR-box fraction cannot uniquely single out quantum theory among generalized nonsignaling theories. | MATH: PR box correlations: P(a,b∣x,y)=12P(a,b|x,y) = \frac{1}{2}P(a,b∣x,y)=21​ if a⊕b=xya \oplus b = xya⊕b=xy, else 0. Bell polytope B\mathcal{B}B: convex hull of local deterministic strategies; its extremal points in (2,2,2) scenario are the 8 local vertices plus 4 PR-box vertices (the "nonlocal" extreme points). The PR box saturates the CHSH inequality: ⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩=4\langle A_0B_0\rangle + \langle A_0B_1\rangle + \langle A_1B_0\rangle - \langle A_1B_1\rangle = 4⟨A0​B0​⟩+⟨A0​B1​⟩+⟨A1​B0​⟩−⟨A1​B1​⟩=4 (Tsirelson bound for quantum: 22≈2.8282\sqrt{2} \approx 2.82822​≈2.828). | CONNECTION: The ratio 22/4=1/2≈0.70712\sqrt{2}/4 = 1/\sqrt{2} \approx 0.707122​/4=1/2​≈0.7071 — not directly in the golden set, but note 0.70712=0.50.7071^2 = 0.50.70712=0.5, and the quantum-to-PR gap 4−22≈1.17164 - 2\sqrt{2} \approx 1.17164−22​≈1.1716 relates to \(2(\sqrt{2}-1) \approx 0.82 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com