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Perfect Strategies for Non-Local Games

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Abstract

We describe the main classes of non-signalling bipartite correlations in terms of states on operator system tensor products. This leads to the introduction of another new class of games, called reflexive games, which are characterised as the hardest non-local games that can be won using a given set of strategies. We provide a characterisation of their perfect strategies in terms of operator system quotients. We introduce a new class of non-local games, called imitation games, in which the players display linked behaviour, and which contain as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and unique games. We associate a C*-algebra \(C^{*}(\mathcal {G})\) to any imitation game \(\mathcal {G}\), and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of \(\mathcal {G}\) can be characterised in terms of properties of this C*-algebra. We single out a subclass of imitation games, which we call mirror games, and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra.

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Acknowledgments

Part of this research was conducted during two Focused Research Meetings, funded by the Heilbronn Institute, and hosted at Queen’s University Belfast in October 2016 and March 2017. M. Lupini was partially supported by the NSF grant DMS-1600186. G. Scarpa acknowledges the support of MTM2014-54240-P (MINECO), QUITEMAD+-CM Reference: S2013/ICE-2801 (Comunidad de Madrid), ICMAT Severo Ochoa project SEV-2015-0554 (MIN-ECO), and grant 48322 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Author notes

  1. M. Lupini

    Present address: School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington, 6140, New Zealand

Authors and Affiliations

  1. Mathematics Department, California Institute of Technology, 1200 East California Boulevard, Mail Code 253-37, Pasadena, CA, 91125, USA

    M. Lupini

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    L. Mančinska

  3. Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

    V. I. Paulsen

  4. Department of Physics, Technical University of Denmark, Richard Petersens Plads, Copenhagen, Denmark

    D. E. Roberson

  5. Universidad Politécnica de Madrid, Escuela Técnica Superior de Ingeniería de Sistemas Informáticos, Calle Alan Turing s/n, 28031, Madrid, Spain

    G. Scarpa

  6. Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK

    S. Severini

  7. Institute of Natural Sciences, Shanghai Jiao Tong University, 200240, Shanghai, China

    S. Severini

  8. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, BT7 1NN, UK

    I. G. Todorov

  9. School of Mathematical Sciences, Nankai University, 300071, Tianjin, China

    I. G. Todorov

  10. ICREA and Física Teòrica: Informació i Fenomens Quàntics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra, Barcelona, Spain

    A. Winter

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  1. M. Lupini
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Lupini, M., Mančinska, L., Paulsen, V.I. et al. Perfect Strategies for Non-Local Games. Math Phys Anal Geom 23, 7 (2020). https://doi.org/10.1007/s11040-020-9331-7

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