Sheaf-Theoretic Preparation Contextuality

TL;DR

This paper introduces a sheaf-theoretic framework for understanding preparation contextuality as an obstruction to stochastic extension, paralleling measurement contextuality but focusing on preparation data.

Contribution

It formalizes preparation contextuality using sheaf theory, identifying structural conditions for admissible extensions and illustrating the concept with a quantum example.

Findings

Preparation contextuality arises from non-extendability of local preparation data.

Structural conditions enforce a rigid product form in extension matrices.

Quantum example demonstrates the presence of preparation contextuality.

Abstract

We introduce a preparation-dual notion of contextuality, formulated as an obstruction to stochastic extension. In parallel with the sheaf-theoretic formulation of measurement contextuality, preparation contextuality arises when locally specified preparation statistics cannot be extended to a single global response matrix compatible with all source contexts. Whereas measurement contextuality concerns the incompatibility of restriction maps (marginalisation), the preparation setting requires stochastic extension of partial conditioning data, which is inherently non-unique. We identify minimal structural and preparation compatibility conditions on admissible extension matrices and show that they enforce a rigid product form. This leads to a notion of preparation contextuality in which the absence of any admissible global response representation witnesses contextuality, while preparation compatibility identifies the cases in which this obstruction is nontrivial. The framework is formulated explicitly in matrix form and illustrated by a quantum-mechanical example exhibiting preparation contextuality.