Linear Algebra of Generalized Contextuality in All Prepare-Transform-Measure Scenarios

2026-07-28

AI Summary

This work addresses the problem of efficiently certifying generalized contextuality in prepare-transform-measure scenarios involving arbitrary sequences of transformations. Building upon a statistics-first linear algebraic framework originally developed for prepare-measure settings, the authors extend it to multi-stage protocols and establish a complete certification procedure. The key contribution lies in the first complete characterization of generalized contextuality for arbitrary sequential transformation scenarios, demonstrating that contextuality can arise solely from the sequential structure of transformations and thereby highlighting the critical role of compositional architecture. The proposed algorithm exhibits exponential complexity in the minimal dimension of the generalized probabilistic theory but only polynomial complexity in the number of processes, and its efficacy is validated through case studies including Spekkens’ toy model and single-qubit stabilizer quantum theory.

Abstract

Generalized contextuality is a canonical distinguishing property of nonclassical generalized probabilistic theories, in particular quantum mechanics. Methods for certification and characterization of generalized contextuality of a given generalized probabilistic theory are well developed for prepare-measure and single-stage prepare-transform-measure scenarios. In a recent work [arXiv:2512.10000], a bottom-up, statistics-first linear-algebraic framework for contextuality in prepare-measure scenarios was introduced. We extend this approach to operational scenarios with sequential transformations with an arbitrary number of stages. We give a full decision procedure for contextuality of such scenarios within operational theories and analyze its computational complexity. In particular, our decision procedure has a complexity linearly exponential in the minimum generalized probabilistic theory (GPT) dimension, and polynomial in the number of procedures. We demonstrate our framework and approach through multiple examples, including Spekkens' toy theory and the 8-state single-qubit stabilizer theory. In particular, we construct an operational theory in which contextuality manifests itself only in the sequential structure of the transformations. Our findings thus shed new light on the significant role of compositional structures in the phenomenon of generalized contextuality.

Problem

• generalized contextuality
• prepare-transform-measure scenarios
• sequential transformations
• operational theories
• generalized probabilistic theories

Innovation

• generalized contextuality
• prepare-transform-measure scenarios
• linear-algebraic framework
• operational theories
• sequential transformations