Full Nonlocality: Extreme Quantum Correlations

• Full nonlocality is defined by vanishing local content, meaning no classical component can account for the observed quantum correlations.
• It extends beyond standard Bell violations by excluding any hybrid models, ensuring that all sources or parties exhibit nonclassical behavior.
• This concept underpins experimental benchmarks and cryptographic protocols by enforcing transitivity and rigorous network-wide nonlocality.

Full nonlocality denotes a family of strongest-possible nonclassicality notions in Bell and network scenarios. In bipartite Bell theory, it commonly refers to correlations with vanishing local content, meaning that no nonzero fraction of the observed statistics admits a local hidden-variable decomposition; in that sense, the correlations are “as nonlocal as any non-signalling correlations” (Aolita et al., 2011). In multipartite and network settings, the phrase is also used for situations in which nonlocality cannot be confined to a subset of parties or sources: certain nonsignalling marginals force other marginals to be nonlocal, and certain network correlations cannot be reproduced unless every source in the network is itself nonlocal (Coretti et al., 2011, Pozas-Kerstjens et al., 2021). More recent work has further linked this extremal regime to geometric faces of the nonsignalling polytope, all-versus-nothing proofs, and pseudotelepathy games, showing that several apparently different notions of maximal nonlocal behaviour coincide under appropriate assumptions (Liu et al., 2023).

1. Conceptual landscape

The basic Bell-theoretic setting is a conditional distribution $P(a,b\mid x,y)$, with local correlations admitting a decomposition of the form

$$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$

while nonsignalling correlations satisfy input-independence of the opposite party’s marginals (Aolita et al., 2011). In this framework, full nonlocality is stricter than ordinary Bell nonlocality: it is not merely the existence of some Bell inequality violation, but the absence of any nonzero local fraction in an Elitzur–Popescu–Rohrlich decomposition

$$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$

with $P_L$ local and $P_{NL}$ nonsignalling. The local content $p_L$ is the maximal achievable $q_L$; full nonlocality corresponds to $p_L=0$ (Aolita et al., 2011).

This Bell-scenario meaning should be distinguished from later network usage. In a network with several independent sources, standard network nonlocality excludes models in which all sources are classical and independent, but it still permits hybrid explanations in which some sources are classical and others are arbitrary nonsignalling resources. Full network nonlocality excludes even those hybrid models: the observed correlations are fully network nonlocal only if no source can be replaced by a local hidden-variable source while preserving the data (Pozas-Kerstjens et al., 2021). A still stronger variant appears in hybrid networks containing multipartite sources, where one rules out not only classical sources but also biseparable nonsignalling ones (Ning-Ning et al., 2024).

A third, related usage concerns unavoidable or propagated nonlocality. In tripartite nonsignalling boxes, one can have pairwise marginals $AB$ and $BC$ such that every nonsignalling completion forces $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$0 to be nonlocal as well. This establishes a transitivity property of nonlocality and motivates a broader reading of “full” nonlocality as a globally enforced structural feature rather than a purely pairwise one (Coretti et al., 2011).

2. Bipartite full nonlocality and local content

The canonical quantitative definition is the local content $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$1 of a bipartite nonsignalling correlation. A correlation is fully nonlocal when $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$2, equivalently when its nonlocal content is $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$3 (Aolita et al., 2011). This notion is stronger than generic Bell inequality violation. For CHSH, the local bound is $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$4, the Tsirelson bound is $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$5, and the nonsignalling bound is $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$6; since the maximal quantum value remains strictly below the nonsignalling bound, even optimal CHSH quantum correlations retain a nonzero local fraction and therefore are not fully nonlocal in this sense (Aolita et al., 2011).

A standard upper bound on $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$7 is obtained from any Bell inequality with local bound $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$8, nonsignalling bound $$P(a,b\mid x,y)=\sum_{\lambda} p(\lambda) P_A(a\mid x,\lambda)\,P_B(b\mid y,\lambda),$$9, and observed value $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$0: $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$1 Hence full nonlocality follows whenever quantum mechanics reaches $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$2, i.e. when the quantum point saturates the nonsignalling bound of the relevant Bell functional (Aolita et al., 2011). The paper “Fully nonlocal quantum correlations” constructs precisely such scenarios by exploiting Kochen–Specker proofs: every Kochen–Specker proof yields a Bell inequality for which the quantum maximum equals the nonsignalling maximum, while the local bound is strictly smaller (Aolita et al., 2011).

The most explicit construction in that work is based on the Peres–Mermin square. It yields a bipartite Bell scenario with three four-outcome measurements per party and Bell functional

$$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$3

For this inequality, the local bound is $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$4, while both the quantum and nonsignalling bounds are $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$5, so the ideal quantum correlation is fully nonlocal (Aolita et al., 2011).

The experimental implementation with hyperentangled photons obtained $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$6, implying $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$7 and, in the authors’ phrasing, providing the strongest reported experimental upper bound on local content in that setting (Aolita et al., 2011). The same work emphasizes that full nonlocality in this sense should not be conflated with generic maximal Bell violation: most familiar inequalities do not admit $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$8, and their optimal quantum points are therefore not fully nonlocal.

3. Geometric, logical, and game-theoretic equivalences

A major conceptual consolidation was achieved by proving the equivalence between four notions: face nonsignalling correlations, full nonlocality, all-versus-nothing proofs, and pseudotelepathy (Liu et al., 2023). In that analysis, a quantum correlation is face nonsignalling if it belongs to a face of the nonsignalling polytope that contains no local points. The same correlation is fully nonlocal if its nonlocal content is $$P(a,b\mid x,y) = q_L P_L(a,b\mid x,y) + (1-q_L)\,P_{NL}(a,b\mid x,y),$$9, i.e. its maximal local weight is zero. The paper proves that these two properties are equivalent for quantum correlations (Liu et al., 2023).

The same work shows that such correlations are exactly those that admit all-versus-nothing proofs. In the bipartite formulation used there, an all-versus-nothing proof is encoded by a table of zero-probability events that is realizable by a quantum correlation but impossible for any local deterministic assignment. This logical contradiction can be turned into a nonlocal game with a perfect quantum strategy and no perfect classical strategy, establishing equivalence with pseudotelepathy (Liu et al., 2023). Conversely, from a perfect pseudotelepathy strategy one recovers a zero-table structure and hence an all-versus-nothing proof.

This equivalence has several consequences. First, full nonlocality need not coincide with maximal violation of a tight Bell inequality. The same paper constructs examples, notably in a pentagram game scenario, of correlations that are fully nonlocal and pseudotelepathic but whose associated Bell inequality is not facet-defining for the local polytope (Liu et al., 2023). This rules out the earlier conjectural intuition that perfect quantum winning strategies should always correspond to maximal violations of tight Bell inequalities.

Second, the equivalence yields a systematic route to proving nonexistence results. The authors introduce a method based on critical nonlocal tables of zeros and semidefinite feasibility checks to decide whether a given Bell scenario can host full-nonlocality/all-versus-nothing/pseudotelepathy correlations (Liu et al., 2023). Using that machinery, they prove that quantum mechanics does not allow such correlations in the $P_L$0 and $P_L$1 Bell scenarios, thereby resolving an open problem whose reformulations span several subfields (Liu et al., 2023).

A plausible implication is that “full nonlocality” is best understood not as an isolated Bell-value extremum but as a polyhedral and logical property of the support of a correlation. The nonsignalling-face viewpoint, the EPR2 decomposition viewpoint, the all-versus-nothing viewpoint, and the pseudotelepathy viewpoint are not merely analogous; in the relevant regime they identify the same resource (Liu et al., 2023).

4. Multipartite propagation and unavoidable nonlocality

In tripartite nonsignalling systems, full or unavoidable nonlocality can appear in a different form: fixed nonlocal marginals may force the remaining marginal to be nonlocal in every compatible nonsignalling completion (Coretti et al., 2011). The operational setting uses boxes $P_L$2 with classical inputs and outputs, with locality defined as convex combinations of deterministic boxes and nonsignalling imposed by linear marginal constraints (Coretti et al., 2011).

The main existence result is a tripartite nonsignalling box with four binary-input, binary-output settings per party such that the $P_L$3 and $P_L$4 marginals both violate the Bell inequality $P_L$5 with value $P_L$6, while the $P_L$7 marginal violates $P_L$8 with value $P_L$9 (Coretti et al., 2011). More importantly, if one fixes the $P_{NL}$0 and $P_{NL}$1 marginals to those values and minimizes the $P_{NL}$2 Bell value over all compatible tripartite nonsignalling boxes, the minimum remains $P_{NL}$3. Therefore every nonsignalling completion necessarily yields nonlocal $P_{NL}$4 correlations (Coretti et al., 2011).

The paper terms this property transitivity of nonlocality. In effect, once the nonlocal $P_{NL}$5 and $P_{NL}$6 behaviours are fixed, nonlocality propagates across the intermediate party $P_{NL}$7; one cannot embed those two marginals into a larger nonsignalling box while keeping $P_{NL}$8 local (Coretti et al., 2011). This provides a distinct sense in which nonlocality becomes “full”: it is not confined to selected edges of a chain but enforced globally by consistency.

The proof method is linear-programming based. The set of nonsignalling boxes is specified by linear equalities and positivity, Bell values are linear functionals, and locality of the $P_{NL}$9 marginal can itself be expressed by linear constraints because the local polytope is the convex hull of local deterministic boxes (Coretti et al., 2011). This makes it possible to optimize Bell expressions over the space of all compatible completions and certify that locality of $p_L$0 is impossible.

The result is especially notable against the backdrop of monogamy. Standard monogamy statements, especially for CHSH, suggest that strong nonlocality between one pair restricts simultaneous nonlocality with another pair. The transitivity construction does not contradict those results because different pairs violate different inequalities, not the same one at the same maximal level; nevertheless it shows that beyond CHSH the structure of multipartite nonsignalling nonlocality is richer than a simple monogamy heuristic suggests (Coretti et al., 2011).

The same paper connects this phenomenon to finite-speed hidden-communication models. If $p_L$1 and $p_L$2 can exchange hidden signals but $p_L$3 cannot because of spacetime arrangement, such models predict local $p_L$4 behaviour; the transitivity construction shows that for suitable nonsignalling marginals this prediction becomes logically inconsistent unless the communication is either observable or effectively infinite in speed (Coretti et al., 2011). The authors explicitly leave open whether their particular transitive correlations are quantum-realizable.

5. Full network nonlocality

Full network nonlocality generalizes Bell nonlocality to networks with independent sources. In the bilocal $p_L$5 scenario, standard bilocal hidden-variable models assume two independent source variables $p_L$6 and decomposition

$p_L$7

(Håkansson et al., 2022). Full network nonlocality is stricter: correlations are fully network nonlocal if they cannot be reproduced by any model in which at least one source is classical while the others are allowed to be arbitrary independent nonsignalling resources (Pozas-Kerstjens et al., 2021). In the bilocal case this means excluding both the C–NS and NS–C hybrid models (Håkansson et al., 2022).

The 2021 formulation established the concept systematically and showed that the standard Branciard bilocal inequality does not witness full network nonlocality, because even its maximal quantum violation can be simulated by a model with one PR-box-type source and one classical source (Pozas-Kerstjens et al., 2021). By contrast, for star networks the generalized inequality

$p_L$8

can detect full network nonlocality, and in the $p_L$9-star case the stronger bound $q_L$0 holds for all non-full-network-nonlocal correlations, while quantum theory reaches $q_L$1 (Pozas-Kerstjens et al., 2021).

A complementary line of work derived explicit hybrid-model witnesses for the bilocal scenario. In the experimental demonstration of full network nonlocality in the bilocal scenario, the relevant inequalities are

$q_L$2

where each witness excludes one placement of the classical source and the other source may be arbitrary nonsignalling (Håkansson et al., 2022). Using two independent SPDC sources, a partial Bell-state measurement, and measurements $q_L$3, $q_L$4, $q_L$5, $q_L$6, the reported values were

$q_L$7

both above the full-network-local bound by more than three standard deviations (Håkansson et al., 2022). The significance is that violating a bilocal inequality alone does not certify end-node entanglement, whereas simultaneous violation of both FNN witnesses excludes all models with one classical source and thus certifies that both links must be nonlocal (Håkansson et al., 2022).

A later photonic experiment closed source-independence, locality, and measurement-independence loopholes while observing full network nonlocality in the same entanglement-swapping architecture (Gu et al., 2023). With two independent sources, fast QRNG-based setting generation, and spacelike separation of relevant events, the measured values at the symmetric Bell-state point were

$q_L$8

exceeding the bound $q_L$9 by more than five standard deviations (Gu et al., 2023). This moved FNN from a source-independence-sensitive laboratory demonstration to a stricter relativistic setting.

The concept also extends beyond bilocal networks. A three-branch photonic star network with three independent sources and a central three-qubit GHZ measurement gave the first experimental demonstration of full network nonlocality beyond the bilocal scenario (Wang et al., 2022). There, one derives inequalities $p_L=0$0, $p_L=0$1, and $p_L=0$2, each excluding models with one specified classical source and the others arbitrary no-signalling independent resources. The reported values

$p_L=0$3

simultaneously violate all three bounds, thereby certifying that all three links are nonclassical without assuming quantum mechanics beyond no-signalling and source independence (Wang et al., 2022).

An even stronger notion, denoted FNN$p_L=0$4, has been proposed for a four-party hybrid network with one tripartite source and one bipartite source (Ning-Ning et al., 2024). Instead of allowing one source to be merely classical, the excluded hybrid model allows the tripartite source to distribute arbitrary biseparable nonsignalling correlations while the bipartite source may be an arbitrary nonsignalling resource. In that setting, the witnesses

$p_L=0$5

simultaneously certify genuine multipartite nonlocality and this stronger full network nonlocality from the same experiment (Ning-Ning et al., 2024). With generalized GHZ and EPR states, the quantum prediction is

$p_L=0$6

which is strictly larger than $p_L=0$7 for all $p_L=0$8 (Ning-Ning et al., 2024).

Recent work has also shown that entangled measurements are not necessary for FNN. In the bilocal scenario, separable measurements at the central node augmented with bidirectional classical feedforward can violate the FNN inequalities

$p_L=0$9

reaching $AB$0 with two singlets (Polino et al., 13 Apr 2026). The same class of measurements can also realize minimal network nonclassicality, a distinct notion in which the correlations are not fully classical but remain compatible with every model having exactly one nonclassical source (Polino et al., 13 Apr 2026). This suggests that measurement entanglement and source nonclassicality are separable resources in network Bell theory.

6. Applications, variants, and open questions

One application area is quantum cryptography. A 2026 study proposed a four-partite entanglement-assisted QKD protocol on a trilocal star network in which security is tied to violation of a trilocal FNN witness and to a corresponding QBER threshold (Mukherjee, 20 Mar 2026). For identical source states, the paper derives a critical error threshold

$AB$1

while for non-identical states it finds

$AB$2

as the threshold associated with the absence of trilocal FNN violation (Mukherjee, 20 Mar 2026). The same work compares this to a CHSH-based multi-link protocol and states that Bell-CHSH-based security tolerates QBER below $AB$3, whereas the FNN-based protocol reduces the threshold below $AB$4, which it interprets as stronger security (Mukherjee, 20 Mar 2026). This suggests that truly network-specific nonlocality may sharpen cryptographic certification relative to pairwise Bell tests.

Another quantitative application is device-independent randomness certification in networks. Using recently developed certification frameworks, the separable-measurement FNN protocol in the bilocal scenario was analyzed against strong and double eavesdropper models (Polino et al., 13 Apr 2026). For the two-party outputs $AB$5, the maximal min-entropy achieved by the separable feedforward strategy was $AB$6 bits against the strong eavesdropper and $AB$7 bits against the double eavesdropper, while entangled-measurement strategies yielded $AB$8 bits and $AB$9 bits respectively (Polino et al., 13 Apr 2026). The implication is that FNN is already sufficient for nontrivial network randomness certification, but the quantitative rate depends strongly on the measurement architecture.

The relation between full nonlocality and quantum realizability remains only partially understood. In the transitivity setting of tripartite nonsignalling boxes, it is explicitly open whether the constructed correlations are quantum (Coretti et al., 2011). In the Bell-scenario equivalence framework, there are complete no-go results for some small input/output scenarios, such as $BC$0 and $BC$1, where quantum theory does not realize any FNS=FN=AVN=PT correlation (Liu et al., 2023). In network settings, even when full NN is proven for certain quantum constructions, the general characterization of all quantum full-network-nonlocal correlations remains open (Pozas-Kerstjens et al., 2021).

The literature also contains uses of “full” nonlocality that are explicitly only heuristic or contrastive. In instantaneous quantum polynomial circuits, for example, nonlocality exists but is hidden: all linear functions of the measurement outcomes satisfy the full-correlation Werner–Wolf–Żukowski–Brukner inequalities, and Bell nonlocality appears only after post-selection or nonlinear processing (Wallman et al., 2014). In that context, “full” nonlocality is not a formal term but refers to the fine-grained, distribution-level structure beyond linear correlators (Wallman et al., 2014). This usage underscores that the phrase can be context-dependent even within quantum information theory.

Taken together, the modern literature supports a plural but coherent picture. In Bell scenarios, full nonlocality is most precisely the vanishing-local-content regime and is equivalent to several extremal logical and polyhedral notions (Aolita et al., 2011, Liu et al., 2023). In multipartite nonsignalling theory, it includes transitive and unavoidable propagation phenomena (Coretti et al., 2011). In networks, it identifies correlations that force every source to be nonclassical, with experimentally demonstrated instances in bilocal, star, and hybrid architectures (Håkansson et al., 2022, Wang et al., 2022, Ning-Ning et al., 2024). The common theme is the exclusion not merely of classical explanations, but of any explanation in which nonclassicality can be localized to only part of the observed structure.