A Combinatorial Approach to Nonlocality and Contextuality

By Antonio Acín, Tobias Fritz, Anthony Leverrier, Ana Belén Sainz. Communications in Mathematical Physics, 2012

205 citations recorded by the source corpus.

AbstractSo 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 Navascués–Pironio–Acín 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 $${G_1}$$G1 and $${G_2}$$G2 with the properties $$\begin{array}{ll}\qquad \qquad \alpha(G_1) \; = \; \Theta(G_1), \qquad \qquad \alpha(G_2) \,= \; \vartheta(G_2), \\ \Theta(G_1\boxtimes G_2) \, > \; \Theta(G_1) \cdot \Theta(G_2),\quad \Theta(G_1 + G_2) \, > \Theta(G_1) + \Theta(G_2).\end{array}$$α(G1)=Θ(G1),α(G2)=ϑ(G2),Θ(G1⊠G2)>Θ(G1)·Θ(G2),Θ(G1+G2)>Θ(G1)+Θ(G2).

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/9bfea81691874205b9c10d106df96a95e2160400

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

  1. Cozmin Ududec
  2. Markus P. Mueller
  3. Howard Barnum

In Atomic and Molecular Physics, and Optics

  1. Cozmin Ududec
  2. Markus P. Mueller
  3. Howard Barnum
  4. S. Hartnoll
  5. Lucien Hardy

In Computational Theory and Mathematics

  1. Cozmin Ududec
  2. Markus P. Mueller
  3. Howard Barnum

Citations in the corpus