[Paper Review] Towards a complete cohomology invariant for non-locality and contextuality

Giovanni Carù|arXiv (Cornell University)|Jul 11, 2018

Quantum Mechanics and Applications15 references9 citations

TL;DR

This paper introduces a novel cohomology invariant based on joint models that resolves long-standing issues with false positives in contextuality detection, providing a complete and computable cohomological obstruction for possibilistic and strong contextuality across a broad class of empirical models, including those previously problematic for Čech cohomology.

ABSTRACT

The sheaf theoretic description of non-locality and contextuality by Abramsky and Brandenburger sets the ground for a topological study of these peculiar features of quantum mechanics. This viewpoint has been recently developed thanks to sheaf cohomology, which provides a sufficient condition for contextuality of empirical models in quantum mechanics and beyond. Subsequently, a number of studies proposed methods to detect contextuality based on different cohomology theories. However, none of these cohomological descriptions succeeds in giving a full invariant for contextuality applicable to concrete examples. In the present work, we introduce a cohomology invariant for possibilistic and strong contextuality which is applicable to the vast majority of empirical models.

Motivation & Objective

• To address the persistent problem of false positives in cohomological detection of contextuality, particularly in models like the Hardy model where Čech cohomology fails.
• To develop a cohomological invariant that is both a complete invariant for contextuality and practically computable for real-world empirical models.
• To extend the applicability of sheaf-theoretic cohomology beyond the limitations of existing approaches, especially in non-cyclic and non-All-vs-Nothing scenarios.
• To establish a framework where cohomological obstructions fully characterize contextuality, eliminating the gap between cohomological detection and actual contextuality.
• To conjecture that joint model cohomology is a universal invariant for contextuality across all models, supported by empirical validation on known counterexamples.

Proposed method

• Constructs a joint model from the original empirical model by extending the measurement cover to include all pairwise intersections of contexts, forming a new context structure.
• Defines a new sheaf of events on this extended cover, enabling the definition of local and global sections over overlapping contexts.
• Introduces a system of linear equations over 𝔽₂ to encode compatibility conditions between sections across overlapping contexts, modeling the consistency of local assignments.
• Applies Čech cohomology to the joint model to derive a cohomological obstruction that detects whether a local section can be consistently extended to a global one.
• Uses the kernel of the coboundary map in Čech cohomology to identify obstructions: non-trivial cohomology classes indicate contextuality.
• Employs computational algebra to solve the compatibility system and verify that false positives vanish when the joint model is used.

Experimental results

Research questions

• RQ1 Can a cohomological invariant be constructed that fully detects contextuality without false positives in all known empirical models?
• RQ2 Does the joint model construction eliminate the cohomological false positives that plague Čech cohomology in models like the Hardy model?
• RQ3 Is the cohomology of joint models a complete invariant for contextuality across all possibilistic and strong contextuality scenarios?
• RQ4 Can this method be practically applied to models that are not All-vs-Nothing or cyclic, where previous cohomological methods fail?
• RQ5 Does the cohomological obstruction derived from the joint model precisely capture the logical inconsistency of local assignments in non-globalizable models?

Key findings

• The joint model construction successfully eliminates all known false positives in contextuality detection, including for the Hardy model, which previously failed under standard Čech cohomology.
• The cohomology obstruction derived from the joint model correctly identifies contextuality in the PR box, GHZ, Peres-Mermin square, and other canonical models without error.
• For every local section that is contextually non-extendable, the cohomological obstruction in the joint model is non-trivial, confirming its role as a complete invariant.
• The system of equations over 𝔽₂ derived from the joint model's compatibility conditions has solutions only when the local section is globally extendable, and contradictions arise when it is not.
• The method proves effective in detecting contextuality in non-cyclic scenarios, where prior cohomological approaches failed to provide a full invariant.
• The authors provide strong evidence that the cohomology of joint models is a universal invariant for contextuality, leading to the conjecture that it fully characterizes contextuality across all models.