Global Kochen-Specker Contextuality Without Local Contextuality and Generalized Bell Nonlocality

Abstract: A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.

• The paper presents evidence that global contextuality arises even when individual subsystems admit local noncontextual and GLHV models.
• It uses three explicit constructions—polarization-path, qubit–qutrit KCBS activation, and a flagged qutrit Werner state—to illustrate the separation between global and local contextuality.
• The work formalizes a compositional criterion, demonstrating that quantum data obstructs a unified global noncontextual description despite locally explainable correlations.

Global Kochen-Specker Contextuality Without Local Contextuality and Generalized Bell Nonlocality

Summary and Main Contributions

The paper presents a rigorous study of global KS-contextuality obstructions arising in multipartite quantum systems for measurement scenarios where all local marginals are noncontextual and all empirically accessible multipartite correlations admit a generalized local hidden-variable (GLHV) description. The work formulates and exhibits scenarios in which quantum models are locally explainable—by both individual subsystem NCHV and scenario-adapted GLHV models—yet no single context-independent global NCHV model exists for the joint data. This property is precisely characterized as “global contextuality,” distinguishing physical (subsystem-based) global/local separation from the context-based definitions used in sheaf-theoretic approaches. The operational implication is that classical compositional explanation is obstructed by quantum theory even in the absence of both Bell and local contextuality violations.

Through three explicit bipartite constructions—a polarization-path scenario, a qubit–qutrit KCBS scenario, and a flagged qutrit Werner-local state—the work separates global contextuality from local contextuality and generalized nonlocality. The constructions include numerical and algebraic analyses of empirical models, explicit enumeration of conditional contextuality, and block-GLHV decomposability proofs. The structural criterion for compositional reduction from GLHV-plus-local-NCHV to GNCHV is formalized and its failure systematically investigated.

Theoretical Framework

The formal setup is an arbitrary multipartite scenario with parties labeled by $i$, each possessing a local measurement set $\mathcal{M}_i$ and a family of compatible measurement contexts $\mathcal{C}_i$. The empirical model is a collection of probabilities over outcomes for each joint (multipartite) context $C = (C_1, \dots, C_n)$ drawn from a prescribed family $$\mathcal{C} \subseteq \mathcal{C}_1 \times \cdots \times \mathcal{C}_n$$. Local noncontextuality (NCHV-local) requires, for each party, a context-independent response function per local observable and a unique hidden variable, per Eq. (1).

GLHV models, generalizing the usual LHV paradigm, associate a unique hidden-variable distribution per tested multipartite context but allow blockwise context-dependent responses within each party (Eq. 2). In contrast, a global noncontextual hidden-variable (GNCHV) model requires a single context-independent response function for all measurements across the system (Eq. 3). The strict hierarchical inclusion—GNCHV ⊆ GLHV ⊆ locally NCHV—underpins the main separation.

Global contextuality, in this framework, is the demonstrated failure to construct a GNCHV model despite the existence of both local NCHV and GLHV models for the accessible data.

Constructions and Analytical Results

1. Polarization-Path GLHV Scenario

The first case studies a bipartite system comprising Alice (a qubit) and Bob (a $2\times4$ polarization-path system with compatible measurement structure $K_{2,2}$). For all $0 < h_0 < 1$, both local and block GLHV models exist for empirical statistics derived from the chosen state and measurement families. However, using a Hardy-type logical contradiction between reported zero events and a positive target event, any GNCHV assignment is ruled out. This is verified by explicit block GLHV linear-feasibility programs that reduce to physical constraints satisfied throughout the relevant parameter region. The contradiction arises not from local or generalized nonlocality, but from the impossibility of gluing context-wise block models into a fully noncontextual description for the system.

2. Qubit–Qutrit KCBS Activation

A qubit–qutrit entangled state is considered, with Bob’s measurements corresponding to a KCBS five-cycle, analyzed via the operator $D$ and KCBS correlator polytope constraints. For all $c_0 \leq c_{\rm KCBS} \simeq 0.850651$, Bob’s unconditional KCBS marginal is noncontextual, and GLHV decomposability is explicitly certified for a reference parameter value ($\mathcal{M}_i$0) by construction of a feasible hidden-variable description. However, conditional on Alice’s measurement yielding a specified outcome, Bob’s postselected marginal achieves maximal KCBS violation. Since local and GLHV constraints do not imply the existence of a global NCHV model, the conditional contextual violation exposes the compositional obstruction: any GNCHV model would make contextual activation by local postselection impossible.

3. Flagged Qutrit Werner-Local State

A state-level example is given by mixing a qutrit Werner-local entangled block with a product flag. For $\mathcal{M}_i$1 (Werner’s projective-local region in $\mathcal{M}_i$2) and with a probability $\mathcal{M}_i$3 for the flag, all local projective statistical models (including all compatible KCBS measurements) are noncontextual. Nevertheless, postselecting on Alice’s flag projects Bob onto a state whose KCBS marginal violates the noncontextuality bound. Since the overall state remains projective-local for all compatible contexts, this constitutes a genuine state-level separation.

Composition Criterion and Structural Implications

A general lemma is proven: if block GLHV responses admit conditional local NCHV factorizations, then the GLHV model can be absorbed into a single GNCHV model by enlarging the hidden-variable space. Conversely, the quantum empirical models for the paper’s constructions violate exactly this gluing criterion—their blockwise factorizability does not extend to a global context-independent assignment.

An explicit “separable reduction” proposition is established: any statistics that can be composed from locally NCHV models for separable states will always admit a GNCHV model. Therefore, the presented obstructions occur strictly in entangled-state regimes where quantum data preclude such reduction.

The negative result—global contextuality without local contextuality or generalized nonlocality—therefore requires certified local NCHV models, verified GLHV certificates for the tested context family, and direct demonstration that no global context-independent assignment is possible, either by conditional activation or logical contradiction.

Implications and Future Directions

The results have several implications for the theory of quantum contextuality and nonlocality:

• Hierarchy of Nonclassicality: The paper completes the logical separation between the absence of local contextuality, the absence of generalized Bell nonlocality, and the existence of a single global noncontextual model. This stresses that empirical nonclassicality is scenario-dependent and may only appear at the compositional level.
• Scenario Design and Certification: The methods used—constructive GLHV decomposition certificates, explicit polytope constraints, and logical witnesses—suggest practical algorithms for searching and certifying new global contextuality scenarios, both for structural (compatibility-induced) and conditional/postselected types.
• Experimental Realization: None of these separations rely on fine-tuned or pathological measurement settings; all data involved can be derived from well-known quantum configurations (Hardy-type paradoxes, KCBS scenarios, Werner states). The explicit state constructions could inform future experimental studies where global contextuality is tested even under the absence of local nonclassicality.
• Resource Theory Considerations: The work distinguishes the compositional obstructions here from LOCC-based resource theories. Rather, the GNCHV/GLHV/NCHV hierarchies are contextual-explanation classes for fixed scenario probabilistic data.
• Further Generalization: The paper suggests that multipartite global contextuality, conditional activation, and compositional analogues of the KS theorem can be systematized using the compositional criterion and blockwise GLHV techniques. Applications may include device-independent protocols or analyses where loophole-free context-gluing is essential.

Conclusion

The paper rigorously demonstrates that global quantum contextuality can be activated in multipartite scenarios where both local contextuality and all forms of Bell-type generalized nonlocality are strictly absent. This provides an unambiguous separation in the classical-explanation hierarchy for quantum correlation data, established through explicit constructive, algebraic, and logical witness arguments. The findings reveal global contextuality as an intrinsic quantum compositional feature and establish a programmatic route for the systematic discovery and certification of such phenomena in quantum information science.