Which Classicality? Incidence, Simplex, and Product-Rule Tests in Finite Quantum Logics

Research Overview

This paper examines the concept of classicality within finite quantum-logical constructions. The author argues that the distinction between classical and quantum behavior is not intrinsic to a system's structure but is instead a consequence of the specific data retained during analysis. By distinguishing between incidence tests (based on valuations and colorings), simplex-embedding tests (based on prepare-and-measure fragments), and operator-functional tests (based on spectral and product rules), the paper clarifies why different tests yield different conclusions for the same physical system.

The Bookkeeping of Classicality

The author categorizes contextuality into three distinct levels:

1. Combinatorial Contextuality: Focuses on the existence of global, context-independent assignments (two-valued states) on an orthogonality hypergraph. This level identifies Kochen-Specker (KS) obstructions.
2. Operational Simplex Contextuality: Examines whether a specific prepare-and-measure table can be embedded into a classical simplex. This is a convex-geometric approach that depends heavily on which preparations and effects are included in the fragment.
3. Operator-Functional Contextuality: Involves algebraic requirements, such as functional-calculus preservation and multiplicative consistency. This layer is crucial for understanding parity arguments like the Greenberger-Horne-Zeilinger (GHZ) proof.

The paper highlights that the GHZ joint measurement is a single Boolean context and only appears nonclassical when one imposes the additional requirement of common local factor assignments and product preservation. By separating these layers, the author provides a framework for understanding why certain constructions, such as two-context pastings, do not inherently produce operational obstructions. This approach allows researchers to identify exactly which constraints—locality, noncontextuality, or product rules—are responsible for the failure of a classical explanation in a given quantum experiment.