The Sheaf-Theoretic Structure Of Non-Locality and Contextuality (1102.0264v7)

Abstract: We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples --- due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively --- occupy successively higher levels of this hierarchy. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling; this is based on a result showing that the linear subspaces generated by the non-contextual and no-signalling models, over an arbitrary measurement cover, coincide. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. A general setting is developed for Kochen-Specker type results, as generic, model-independent proofs of maximal contextuality, and a new combinatorial condition is given, which generalizes the `parity proofs' commonly found in the literature. We also show how our abstract setting can be represented in quantum mechanics. This leads to a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.

• The paper introduces a sheaf-theoretic approach that unifies the study of non-locality and contextuality in quantum mechanics.
• It establishes a hierarchy distinguishing probabilistic, possibilistic, and strong contextuality, clarifying their roles in models like Bell, Hardy, and GHZ.
• It applies linear algebra to reveal obstructions to global sections, linking negative probabilities to the no-signalling condition and CSP unsatisfiability.

Sheaf-Theoretic Structure Of Non-Locality and Contextuality

This paper explores the intricate connections between non-locality and contextuality in quantum mechanics using a sheaf-theoretic framework. Non-locality and contextuality, fundamental features of quantum theory, defy classical intuitions and appear prominently in no-go theorems such as Bell's theorem and the Kochen-Specker theorem. The authors propose a unified mathematical framework based on sheaf theory that generalizes the traditional probability tables used in non-locality theory to arbitrary measurement covers.

Key Contributions

1. Unified Framework: The paper introduces a sheaf-theoretic approach to examining non-locality and contextuality. This methodology allows for the treatment of both concepts in a unified setting, demonstrating that non-locality can be perceived as a specific form of contextuality. This approach enables proofs in considerable generality, independent of traditional Hilbert space methods.
2. Hierarchy of Non-Locality: The paper distinguishes among various degrees of non-locality—standard probabilistic, possibilistic, and strong contextuality. These form a hierarchy, with strong contextuality implying possibilistic non-locality, and possibilistic non-locality implying probabilistic non-locality. Notably, prominent quantum models, such as those of Bell, Hardy, and GHZ, occupy ascending levels within this hierarchy.
3. Linear Algebra and Obstructions: The authors describe a linear algebraic method for identifying obstructions to global sections, characterizing non-locality and contextuality precisely. This method uncovers new insights into no-go theorems and allows a systematic treatment of these concepts.
4. Negative Probabilities and No-Signalling: The paper shows that the existence of local hidden-variable realizations with negative probabilities is equivalent to the no-signalling condition. This connection highlights that negative probabilities cannot solely characterize quantum mechanics, contrary to some suggestions in the literature.
5. Strong Contextuality and CSPs: The concept of strong contextuality is explored, establishing that it characterizes models with no global sections even for supports. A connection is made between maximal contextuality and CSPs, generalizing maximal non-locality as unsatisfiability in constraint satisfaction.

Implications and Future Work

The implications of this work are both profound and far-reaching. The sheaf-theoretic framework allows for the application of extensive mathematical techniques, such as cohomology, to explore non-locality and contextuality. The characterization of these phenomena through global sections offers a mathematically robust way of understanding these quantum properties, potentially leading to new theorems and insights.

In practice, this can influence experimental setups and interpretations in quantum mechanics, offering a more comprehensive understanding of the limitations and capabilities of quantum systems. The speculative extension of these ideas might also intersect with computational fields, especially in understanding complex systems and algorithms that exhibit contextual or non-local behaviors.

Conclusion

The paper presents a sophisticated sheaf-theoretic approach that mathematically formalizes non-locality and contextuality in quantum mechanics. By extending traditional frameworks and introducing new hierarchies and characterizations, the paper provides a broader understanding of quantum phenomena. Its contributions are poised to enrich both theoretical inquiries and experimental applications in quantum mechanics, setting the stage for further advancements in the understanding of quantum theory.