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A Combinatorial Approach to Nonlocality and Contextuality

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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 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 \({G_1}\) and \({G_2}\) with the properties

$$\begin{array}{ll}\qquad \qquad \alpha(G_1) \; = \; \Theta(G_1), \qquad \qquad \alpha(G_2) \,= \; \vartheta(G_2), \\ \Theta(G_1\boxtimes G_2) \, > \; \Theta(G_1) \cdot \Theta(G_2),\quad \Theta(G_1 + G_2) \, > \Theta(G_1) + \Theta(G_2).\end{array}$$

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Authors and Affiliations

  1. ICFO-Institut de Ciències Fotòniques, 08860, Castelldefels, Barcelona, Spain

    Antonio Acín & Ana Belén Sainz

  2. ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010, Barcelona, Spain

    Antonio Acín

  3. Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada

    Tobias Fritz

  4. INRIA Rocquencourt, Domaine de Voluceau, B.P. 105, 78153, Le Chesnay Cedex, France

    Anthony Leverrier

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  1. Antonio Acín
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  4. Ana Belén Sainz
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Correspondence to Tobias Fritz.

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Communicated by A. Winter

We thank Mateus Araújo, Adán Cabello, Ravi Kunjwal, Simone Severini, Alexander Wilce, Andreas Winter, Elie Wolfe and Gilles Zémor for comments and discussion, András Salamon for help with a reference, and Will Traves for help on \({\texttt{MathOverflow}}\). Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. T.F. was supported by the John Templeton foundation. Part of this work was done while A.L. was at the Institute for Theoretical Physics, ETH Zürich. A.L. was supported by the Swiss National Science Foundation through the National Centre of Competence in Research “Quantum Science and Technology”, by the CNRS through the PEPS ICQ2013 TOCQ, and through the European Research Council (Grant No. 258932). A.B.S. was supported by the ERC SG PERCENT and by the Spanish projects FIS2010-14830 and FPU:AP2009-1174 PhD grant. A.A. was supported by the ERC CoG QITBOX, the Spanish projects FOQUS and DIQIP and the John Templeton Foundation.

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Acín, A., Fritz, T., Leverrier, A. et al. A Combinatorial Approach to Nonlocality and Contextuality. Commun. Math. Phys. 334, 533–628 (2015). https://doi.org/10.1007/s00220-014-2260-1

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