A Combinatorial Approach to Nonlocality and Contextuality (1212.4084v5)

Abstract: So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we also apply to contextuality scenarios. We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant. In terms of graph theory, our main result is this: there exist two graphs $G_1$ and $G_2$ with the properties \begin{align*} \alpha(G_1) &= \Theta(G_1), & \alpha(G_2) &= \vartheta(G_2), \[6pt] \Theta(G_1\boxtimes G_2) & > \Theta(G_1)\cdot \Theta(G_2),& \Theta(G_1 + G_2) & > \Theta(G_1) + \Theta(G_2). \end{align*}

• The paper presents a hypergraph framework that refines previous methods by incorporating explicit probability normalization constraints.
• It establishes strong links between contextuality scenarios and graph invariants, bridging classical, quantum, and generalized probabilistic models.
• The study applies an NPA hierarchy to Bell scenarios, offering new insights into quantum nonlocality and raising open problems for future research.

A Combinatorial Approach to Nonlocality and Contextuality

The paper "A Combinatorial Approach to Nonlocality and Contextuality" by Antonio Acín and collaborators develops a comprehensive formalism to understand quantum contextuality scenarios using the combinatorial structure of hypergraphs. This approach refines previous works by explicitly including probability normalization constraints and extends the applicability to several probabilistic models, such as classical, quantum, and generalized models. It is, therefore, a significant contribution to the understanding of nonlocality and contextuality in quantum mechanics.

Key Contributions

1. Hypergraph Framework: The paper proposes a hypergraph-based framework for contextuality scenarios, significantly refining the previous method introduced by Cabello, Severini, and Winter (CSW). The framework includes normalization of probabilities, which allows for stricter control over probabilistic model sets.
2. Connection to Graph Theory: This combinatorial approach establishes strong links between contextuality scenarios and various graph invariants. The relationship between contextuality scenarios and graph structures like independence number, Lovász number, and Shannon capacity are explored in detail.
3. Local Orthogonality Principle: The paper highlights the recently proposed Local Orthogonality principle as a specific instance of a general principle related to the Shannon capacity of graphs. It is shown that this principle is not maximal but can be dominated by a level in the Navascués-Pironio-Acín (NPA) hierarchy of semidefinite programs, which is also applied to contextuality scenarios.
4. Characterization and Open Problems: The authors derive various results linking quantum and supraquantum nonlocality, contextuality, and provide numerous open problems. They argue, notably, that quantum models in a scenario cannot generally be characterized by a graph invariant.

Results and Discussion

• Graph Invariants: The paper presents results concerning graph products (such as strong and disjoint union products) and their impact on graph invariants, demonstrating that the inequalities for Shannon capacities are not always tight near the quantum limit.
• Hierarchies and Relaxations: By incorporating a methodology similar to the NPA hierarchy, the authors provide a pathway to approximate contextuality scenarios using a sequence of semidefinite programs, which gradually converge to the quantum set.
• Implications for Nonlocality: For the case of Bell scenarios, which are derived from products of contextuality scenarios, the framework describes how conventional notions of nonlocality arise naturally, updated to include the normalization constraints that are typically absent in pre-CSW treatments.

Conclusion and Future Developments

This paper presents a substantial step toward a richer mathematical treatment of quantum contextuality and nonlocality, proposing a robust formalism to explore these phenomena via a combinatorial lens. By doing so, it opens pathways for new theoretical insights and raises several questions regarding the structure and bounds of quantum mechanics through the lens of hypergraph theory.

In terms of future work, the results hint at the potential for discovering further connections between physical principles and graph-theoretic constructs, which might lead to new characterizations or constraints applicable to quantum mechanics. There is also an opportunity to explore the activation of Consistent Exclusivity violations further, as well as to explore the relationship between probabilistic and quantum graph invariants. This combinatorial approach forms a basis from which many future theoretical and computational developments can emerge, potentially influencing advancements in quantum information and quantum computing.