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

Partha Ghose

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.

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

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. ). 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.
Paper Structure (19 sections, 3 theorems, 20 equations, 2 figures)

This paper contains 19 sections, 3 theorems, 20 equations, 2 figures.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Operational classicality ("classical" in the present sense):
  4. Polarization systems and an entangling preparation
  5. Two-beam polarization systems and a fixed tensor order
  6. Idealized target: Hadamard then CNOT
  7. Alternative preparation via external conical refraction (ECR)
  8. Process layer: $\mathbf{CPM}(\mathbf{FHilb})$ and conditioning
  9. Ambient category and operational primitives
  10. Preparation as a single morphism with a flag system
  11. Interpretation of the flag system $E$:
  12. Local measurement settings and correlations
  13. Sheaf layer: contextuality as failure of gluing
  14. CHSH: product states glue; Bell-like outputs can fail to glue
  15. Discussion and conclusion
  16. ...and 4 more sections

Key Result

Lemma 5.1

If the empirical model admits a global section (equivalently, a joint distribution on $O^X$), then $|S|\le 2$.

Figures (2)

  • Figure 1: Bell-state preparation in two complementary views. (a) Abstract circuit: a Hadamard on one input followed by a CNOT produces a Bell state. (b) Optical realisation (schematic): a Hadamard-like splitting and a CNOT-like routing/flip, implemented with standard linear-optical elements and subsequent recombination/conditioning, prepares a Bell-like two-beam state. In practice one uses coherent laser modes (for phase stability), while polarization stochasticity is introduced by controlled modulation/scrambling of the Jones/Stokes variables; elements such as VP are then understood as state-preparation (projection/conditioning) steps rather than unitary gates.
  • Figure 2: Schematic comparison: intersecting SPDC emission cones (left) and an ECR-based analogue (right), where intersecting conical-refraction ring sectors and subsequent routing/conditioning select a Bell-like two-beam polarization branch.

Theorems & Definitions (7)

  • Remark 2.1: Physical "term elimination" is part of the preparation
  • Definition 3.1: Conditioned (postselected) output
  • Definition 4.1: Empirical model AbramskyBrandenburger2011
  • Definition 4.2: Noncontextuality as a global section AbramskyBrandenburger2011
  • Lemma 5.1: CHSH bound from global sections
  • Proposition 5.1: Product states glue
  • Proposition 5.2: CHSH violation $\Rightarrow$ no global section