Global Contextuality in Quantum Systems

• Global contextuality is a property of empirical models where locally consistent probability distributions fail to extend to a single classical global description.
• It is formalized using sheaf-theoretic and bundle frameworks that expose obstructions, such as the nonexistence of global sections, to classical representability.
• This concept underpins phenomena like Bell nonlocality and informs resource theories by quantifying contextual behavior using topological and homological methods.

Searching arXiv for recent and foundational papers on global contextuality to ground the article. Global contextuality is the property of an empirical model whereby locally compatible statistical data cannot be extended to a single global classical description. In the Abramsky–Brandenburger sheaf-theoretic formulation, and in closely related bundle formulations, this means that a family of context-wise probability distributions admits no global section: there is no joint probability distribution over hypothetical outcomes of all measurements whose marginals reproduce the observed distributions in every context (Abramsky et al., 2011). In the bundle formulation, the same point is expressed as “the impossibility to construct a globally consistent description of a model even if there is local agreement,” with contextuality identified as the non-existence of a global section in the measure bundle (Montanhano, 2021).

1. Definition, scope, and terminological distinctions

A measurement scenario consists of a set of measurements and a family of contexts, where each context is a jointly performable subset. For each context $C$, an empirical model specifies a joint probability distribution $e_C$, and local compatibility requires agreement of marginals on overlaps: if $U \subseteq C,C'$, then $e_C|_U = e_{C'}|_U$. Global noncontextuality is the existence of a single distribution

$d : O^X \to [0,1]$

such that $d|_C = e_C$ for every context $C$; global contextuality is the failure of such a distribution to exist (Abramsky et al., 2011).

This notion unifies several standard no-go phenomena. Bell nonlocality is a special case of contextuality in scenarios whose contexts are fixed by spatial separation, and Leggett–Garg-type temporal inequalities rest on the same underlying assumption: the existence of a joint probability distribution for all relevant observables or times (Markiewicz et al., 2013). A recurring theme across these frameworks is that contextuality is not a defect of a single context; it is a system-level obstruction to gluing all contexts into one classical model.

A persistent source of confusion is the distinction between global contextuality and Bell contextuality. Griffiths reserves “Bell contextuality” for counterfactual claims about whether the outcome of $A$ would change when measured with $B$ rather than with $C$, whereas global contextuality concerns whether one can construct a single joint probability distribution over all observables, compatible or incompatible (Griffiths, 2019). A different terminological variant appears in recent multipartite work, where “global” refers to the whole composite system rather than to the sheaf-theoretic passage from local contexts to a global section; in that usage, global contextuality is a compositional obstruction to a single noncontextual hidden-variable model for the whole multipartite experiment (Yang, 27 May 2026).

2. Sheaf-theoretic and bundle formulations

Abramsky and Brandenburger formulate contextuality in terms of the event sheaf $e_C$0, with $e_C$1 for each set $e_C$2 of measurements and restriction maps given by projection. An empirical model is a compatible family $e_C$3 in the presheaf $e_C$4 of distributions on sections, and contextuality is exactly the obstruction to a global section $e_C$5 satisfying $e_C$6 for all maximal contexts (Abramsky et al., 2011).

The fibre-bundle approach recasts the same idea geometrically. In “On Measures and Measurements,” the local data of a non-disturbing empirical model define a probability fibre bundle: each context $e_C$7 carries a probability space $e_C$8, and overlaps are glued by identification of the corresponding sub-$e_C$9-algebras together with equality of marginal measures. Noncontextuality is then the existence of a trivial probability bundle, equivalently a single global probability space whose marginals recover all local models; contextuality is the non-triviality of that bundle (Cunha, 2019).

The later bundle treatment of contextuality as simplicial-complex topology sharpens this picture. There a scenario $U \subseteq C,C'$0 is treated as a simplicial complex, with contexts as simplices, and contextuality is encoded in a measurable or measure bundle over that complex. At the level of measures, global noncontextuality is the existence of a measure $U \subseteq C,C'$1 on the full outcome space $U \subseteq C,C'$2 such that

$U \subseteq C,C'$3

whereas global contextuality is the non-existence of such a global measure (Montanhano, 2021).

Kishida’s logic of local inference addresses the complementary half of the same structure. In that setting, contextuality is “global inconsistency” together with “local consistency”: local sections exist and are compatible, yet no global section exists. The inchworm logic formalizes what follows locally in no-signalling presheaf models, while the failure of global sections remains the mark of contextuality (Kishida, 2016).

3. Equivalent classical representations and extension theorems

A central result is the Fine–Abramsky–Brandenburger theorem. In the bundle formulation with measurable outcome spaces, an empirical model is noncontextual if and only if three conditions are equivalent: it is noncontextual by model, meaning that it admits a factorizable hidden-variable model; it is noncontextual by marginals, meaning that it admits a global measure $U \subseteq C,C'$4; and it is noncontextual by section, meaning that every local section extends to a global section (Montanhano, 2021). In the original sheaf-theoretic formulation, the existence of a global section is likewise equivalent to the existence of a factorizable hidden-variable realization (Abramsky et al., 2011).

The hidden-variable form is the familiar classical factorization condition. For a context $U \subseteq C,C'$5, one writes

$U \subseteq C,C'$6

with deterministic response measures on minimal contexts. The theorem identifies this with the existence of a single global probability model on all measurements (Montanhano, 2021).

The continuous-variable extension preserves this structure. In continuous-variable scenarios, the event sheaf is built from measurable spaces, the probability presheaf uses the Giry construction, and the Fine–Abramsky–Brandenburger theorem continues to hold: extendability to a global probability measure, realization by a deterministic hidden-variable model, and realization by a factorisable hidden-variable model remain equivalent (Barbosa et al., 2019). This is important because it shows that Bell nonlocality remains a special case of contextuality beyond finite outcome sets.

4. Topology, cohomology, homology, and holonomy

Once contextuality is identified with failure of global extension, topology becomes a natural refinement. In the cohomological program, obstructions to global sections are represented by cohomology classes. Mansfield and collaborators showed that these cohomological obstructions are not a complete invariant for strong contextuality, even under symmetry and connectedness restrictions on the measurement cover, thereby disproving an earlier conjecture (Carù, 2017). The same work generalized the obstruction theory to higher cohomology groups and organized different “levels” of contextuality into a hierarchy of logical implications.

The bundle approach develops a closely related homological refinement through $U \subseteq C,C'$7-contextuality. If $U \subseteq C,C'$8 is the contextual fraction of the model restricted to the $U \subseteq C,C'$9-skeleton of the scenario, then

$e_C|_U = e_{C'}|_U$0

This isolates the amount of contextuality that first appears when $e_C|_U = e_{C'}|_U$1-dimensional simplices are included (Montanhano, 2021). The tetrahedron model shows that contextuality can be invisible on the $e_C|_U = e_{C'}|_U$2-skeleton and appear only at dimension $e_C|_U = e_{C'}|_U$3: in that example $e_C|_U = e_{C'}|_U$4 while $e_C|_U = e_{C'}|_U$5, so the model is $e_C|_U = e_{C'}|_U$6-contextual (Montanhano, 2021). GHZ models realize all levels of the hierarchy: for the $e_C|_U = e_{C'}|_U$7-partite GHZ scenario, all lower-dimensional marginals are noncontextual,

$e_C|_U = e_{C'}|_U$8

while at the full dimension $e_C|_U = e_{C'}|_U$9 (Montanhano, 2021).

The same paper introduces a geometric treatment of $d : O^X \to [0,1]$0-contextuality through connections and holonomy. For graph scenarios, Markov kernels along edges define a connection on the measure bundle; when outcome fibres are equal, these kernels induce orthogonal transformations on a frame bundle. If the empirical model is $d : O^X \to [0,1]$1-noncontextual and has no singular paths, then the associated holonomy group is trivial. Non-trivial holonomy therefore implies contextuality or singularity, while trivial holonomy is necessary but not sufficient for $d : O^X \to [0,1]$2-noncontextuality (Montanhano, 2021). This gives a discrete geometric picture in which local transport is well defined but global flatness fails.

5. Quantification and resource-theoretic structure

Quantitative treatments of global contextuality begin with the idea that noncontextuality may fail by degree rather than merely by kind. Svozil proposed measuring contextuality by the probability with which context-dependent violations must occur in a distribution over global assignments of values to context-labelled observables (Svozil, 2011). In the CHSH scenario, the algebraic maximum $d : O^X \to [0,1]$3 requires maximal contextuality, while Tsirelson’s bound $d : O^X \to [0,1]$4 requires contextual-to-noncontextual assignments in the ratio

$d : O^X \to [0,1]$5

for the chosen observable (Svozil, 2011). This is explicitly a global notion because the probability distribution ranges over the entire space of global assignments.

The contextual fraction later became the standard global quantifier. In the sheaf-theoretic formulation, it is defined by a convex decomposition

$d : O^X \to [0,1]$6

where $d : O^X \to [0,1]$7 is noncontextual, $d : O^X \to [0,1]$8 is strongly contextual, $d : O^X \to [0,1]$9, and $d|_C = e_C$0 (LeMaitre, 15 Aug 2025). In continuous-variable scenarios, $d|_C = e_C$1 is defined through subprobability global models and becomes an infinite linear program, which can be approximated by a hierarchy of semidefinite programs via Lasserre relaxations (Barbosa et al., 2019).

The resource-theoretic treatment takes globally noncontextual behaviors as free objects and behaviors without global sections as resources. In the resource theory based on noncontextual wirings, the free operations of a given type form a polytope whose vertices are deterministic noncontextual wirings, and the preorder of resource convertibility is locally infinite, not totally ordered, not weak, and of infinite height and width (Santos et al., 2022). Cost and yield monotones based on $d|_C = e_C$2-cycle inequalities generalize earlier Bell-nonlocality results to arbitrary compatibility scenarios (Santos et al., 2022).

Recent relativistic work uses the same global quantifier operationally. In contextuality harvesting, detectors initially prepared in noncontextual states interact locally with a quantum field vacuum, and the post-interaction detector statistics are contextual precisely when their empirical model admits no global noncontextual distribution. The harvested contextuality is quantified by

$d|_C = e_C$3

and the contextual fraction is presented as a unified measure for local Kochen–Specker contextuality, Bell-type entanglement, and magic (LeMaitre, 15 Aug 2025).

6. Extensions, alternative frameworks, and interpretational issues

One important extension concerns signalling and inconsistent connectedness. The original sheaf-theoretic framework assumes compatibility on overlaps, whereas Contextuality-by-Default treats measurements of the same content in different contexts as distinct random variables and defines noncontextuality via multimaximal couplings. For cyclic systems with binary outcomes, this yields a necessary and sufficient criterion

$d|_C = e_C$4

which reduces to CHSH, KCBS, and Leggett–Garg inequalities under consistent connectedness (Dzhafarov et al., 2015). Dzhafarov later showed that, after consistification, the sheaf-theoretic and Contextuality-by-Default approaches become essentially coextensive (Dzhafarov, 2019).

A second extension is preparation contextuality. The preparation-dual framework defines preparation contextuality as an obstruction to stochastic extension: locally specified preparation statistics fail to extend to a single global response matrix compatible with all source contexts. Under input independence and compositionality, admissible extension matrices are forced into a rigid product form, and preparation compatibility becomes the analogue of no-signalling (Williams et al., 1 May 2026). In that setting, the absence of any admissible global response representation witnesses preparation contextuality.

Interpretationally, global contextuality is a property of empirical models, not a direct statement that incompatible observables possess jointly meaningful values. Griffiths emphasized that a mathematically well-defined global distribution can exist even when it has no direct quantum meaning because incompatible observables do not define a common quantum sample space (Griffiths, 2019). This caution does not negate the formal notion; rather, it clarifies that global contextuality measures the limits of classical representability of context-wise statistics. A related recent development shows that, in multipartite scenarios with arbitrary local compatible contexts and no cross-party joint measurements, even the conjunction of local Kochen–Specker noncontextuality and generalized Bell locality need not imply a single global noncontextual hidden-variable model for the whole system (Yang, 27 May 2026).

Taken together, these lines of work define global contextuality as a structural obstruction to classical extension. Whether expressed as the failure of a global section, the non-triviality of a probability or measure bundle, the nonexistence of a global response matrix, or the impossibility of a multimaximally connected coupling, the common content is the same: local consistency does not guarantee global classicality.