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arXiv:2602.14508

Bell-like States in Classical Optics: A Process-Theoretic and Sheaf-Theoretic (Categorical) Clarification

Partha Ghose

quant-phSubmitted Feb 16, 2026• Announced Feb 17, 2026• 0 citations
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Abstract

Classical polarization optics is naturally described by a two-dimensional complex Hilbert space (Jones vectors), so the tensor-product kinematics underlying bipartite nonseparability is already available classically. For statistical (stochastic) optical fields, and under an operational stance where outcomes are not assumed pre-assigned prior to detection, suitably prepared two-beam polarization states can exhibit Bell--CHSH correlations of quantum strength. The same platform offers a tunable, low-cost testbed for stress-testing Bell/CHSH and contextuality witnesses under realistic imperfections (noise, coarse binning, selective sampling). We also outline an alternative preparation based on external conical refraction (ECR), where engineered intersecting conical-refraction rings mimic the intersecting emission cones of SPDC. We give a self-contained categorical formulation: the preparation-and-conditioning pipeline (Hadamard-like splitting, CNOT-like coupling, and routing/conditioning that removes unwanted contributions) is treated as a single morphism in an operational process theory (e.g. $\mathbf{CPM}(\mathbf{FHilb})$). From it we functorially extract an empirical model, i.e. a compatible family of context-indexed probability distributions. The Abramsky--Brandenburger sheaf criterion then applies: noncontextuality is the existence of a global section, and CHSH violation is a precise failure-to-glue. This separates kinematic nonseparability from operational contextuality and clarifies why neither, by itself, entails nonlocal causation; contextuality can arise in a classically implementable stochastic-optics regime.

Comments
12 pages, two figures and two line drawings

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Classical polarization optics is naturally described by a two-dimensional complex Hilbert space (Jones vectors), so the tensor-product kinematics underlying bipartite nonseparability is already available classically. For statistical (stochastic) optical fields, and under an operational stance where outcomes are not assumed pre-assigned prior to detection, suitably prepared two-beam polarization states can exhibit Bell--CHSH correlations of quantum strength. The same platform offers a tunable, low-cost testbed for stress-testing Bell/CHSH and contextuality witnesses under realistic imperfections (noise, coarse binning, selective sampling). We also outline an alternative preparation based on external conical refraction (ECR), where engineered intersecting conical-refraction rings mimic the intersecting emission cones of SPDC. We give a self-contained categorical formulation: the preparation-and-conditioning pipeline (Hadamard-like splitting, CNOT-like coupling, and routing/conditioning that removes unwanted contributions) is treated as a single morphism in an operational process theory (e.g. CPM(FHilb)\mathbf{CPM}(\mathbf{FHilb})CPM(FHilb)). From it we functorially extract an empirical model, i.e. a compatible family of context-indexed probability distributions. The Abramsky--Brandenburger sheaf criterion then applies: noncontextuality is the existence of a global section, and CHSH violation is a precise failure-to-glue. This separates kinematic nonseparability from operational contextuality and clarifies why neither, by itself, entails nonlocal causation; contextuality can arise in a classically implementable stochastic-optics regime.
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.
A method is proposed to produce a classical optical state that is `intersystem nonseparable' and a close analog of the ϕ+ϕ^+ϕ+ Bell state. A derivation of the CHSH-Bell inequality is sketched within the framework of classical polarization optics, using non-contextuality for factorizable states as an axiom rather than any hidden variable theory. It is shown that the classical state violates this inequality.
A novel approach for analyzing "classical" alternatives to quantum mechanics for explaining the statistical results of an EPRB-like experiment is proposed. This perspective is top-down instead of bottom-up. Rather than beginning with an inequality derivation, a hierarchy of model types is constructed, each distinguished by appropriately parameterized conditional probabilities. This hierarchy ranks the "classical" model types in terms of their ability to reproduce QM statistics or not. The analysis goes beyond the usual consideration of model types that "fall short" (i.e., satisfy all of the CHSH inequalities) to ones that are "excessive" (i.e., not only violate CHSH but even exceed a Tsirelson bound). This approach clearly shows that noncontextuality is the most general property of an operational model that blocks replication of at least some QM statistical predictions. Factorizability is naturally revealed to be a special case of noncontextuality. The same is true for the combination of remote context independence and outcome determinism (RCI+OD). It is noncontextuality that determines the dividing line between "classical" model instances that satisfy the CHSH inequalities and those that don't. Outcome deterministic operational models are revealed to be the "building blocks" of all the rest, including quantum mechanical, noncontextual, and contextual ones. The set of noncontextual model instances is exactly the convex hull of all 16 RCI+OD model instances, and furthermore, the set of all model instances, including all QM ones, is equal to the convex hull of the 256 OD model instances. It is shown that, under a mild assumption, the construction of convex hulls of finite ensembles of OD model instances is (mathematically) equivalent to the traditional hidden variables approach. Plots and figures provide visual affirmation of many of the results.
Stochastic optics may be considered as simply a local realistic interpretation of quantum optics and, in this sense, it is a first step in the reinterpretation of the whole of quantum theory. However, as it is not possible to interpret all the details of quantum theory in a local realistic manner, as shown by Bell's theorem, minor changes are introduced in the formalism with the consequence that the new theory makes different predictions in some special cases. In stochastic optics, the quantum-operator formalism is simply considered a formal way of dealing with stochastic fields. In particular, the quantum zero point is taken as a real random electromagnetic radiation filling the whole of space. This radiation noise has the same nature as light signals, the only difference being the greater intensity of the latter. We assume that photon detectors have an intensity threshold just above the level of the noise, thus detecting only signals. Transmission of radiation through polarizers follows Malus's law, but the interplay of signal and noise leads quite naturally to the prediction that the detection probability of some signals is enhanced, which is known to be a necessary condition for the violation of the empirically tested Bell inequalities. In our view, correlated photon pairs are pairs of light signals supercorrelated in polarization, in the sense that, as well as the signal, the accompanying noise is also correlated. Thus stochastic optics allows predictions for the empirical correlations very close, but not identical, to the quantum ones. The theory is applied to the analysis of all experiments designed to test the Bell inequalities by measuring polarization correlations of photon pairs. The predictions agree with quantum optics and experiments within statistical errors, except for the Holt-Pipkin experiment. In this case, the experimental results agree with stochastic optical predictions within two standard deviations while violating quantum optics by four.
Since Bell, Kochen and Specker formulated the first notions of contextuality, a number of frameworks have been created to help make philosophical and mathematical sense of the concept. Two of the most common are the Equivalence-based approach due to Spekkens, and the Sheaf-theoretic framework of Abramsky and Brandenburger. Each of these can support a hierarchy of strengths of contextuality, which can include possibilistic concepts. This thesis will explore the differences between the two approaches, and explore what can be demonstrated by using weaker notions of noncontextuality. Notions of noncontextuality are important for proving no-go theorems which highlight that many properties that are true of classical mechanics fail to be compatible with quantum predictions. Often, the notions of noncontextuality used are very strong assumptions, and even weaker flavours of noncontextuality can be used to draw the same, or similar conclusions. An important class of such notions are, possibilistic assumptions that only care about whether or not events are possible or impossible, rather than the exact Born rule probabilities that are applied to events. Even these weaker notions are often impossible to resolve with the logical structure and predictions of quantum theory. Contextuality has been demonstrated to be a resource for many quantum computational operations, and the contextual fraction has been shown to be an important resource. We introduce a metric that can differentiate between different quantum mechanical scenarios with contextual fraction 1, and demonstrate analogues of CHSH and Tsirelson bounds for this quantity. We also investigate the computational complexity of calculating the values of some metrics for nonlocality and contextuality, completing the classification of possibilistic nonlocality scenarios, as well as other metrics of nonlocality analogous to our metric of maximal contextuality.