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Linear conic formulations for two-party correlations and values of nonlocal games

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Abstract

In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the Navascués, Pironio and Acín (NPA) hierarchy and also a sufficient condition for the set of quantum correlations to be closed. Furthermore, by our conic formulations, the value of a nonlocal game over the sets of classical, quantum, no-signaling and unrestricted correlations can be cast as a linear conic program. This allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lovász is in fact an upper bound to the quantum value of the game and moreover, it is at least as strong as optimizing over the first level of the NPA hierarchy. Lastly, we show that deciding the existence of a perfect quantum (resp. classical) strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices (resp. completely positive matrices). By specializing the results to synchronous nonlocal games, we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature.

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Acknowledgments

The authors would like to thank the referees for carefully reading the paper and for their useful comments. Furthermore, the authors thank S. Burgdorf, M. Laurent, L. Mančinska, T. Piovesan, D. E. Roberson, S. Upadhyay, T. Vidick and Z. Wei for useful discussions. A.V. is supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. J.S. is supported in part by NSERC Canada. Research at the Centre for Quantum Technologies at the National University of Singapore is partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant “Random numbers from quantum processes,” (MOE2012-T3-1-009).

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Authors and Affiliations

  1. Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

    Jamie Sikora & Antonios Varvitsiotis

  2. MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore, Singapore

    Jamie Sikora

  3. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Antonios Varvitsiotis

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  1. Jamie Sikora
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  2. Antonios Varvitsiotis
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Correspondence to Antonios Varvitsiotis.

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Sikora, J., Varvitsiotis, A. Linear conic formulations for two-party correlations and values of nonlocal games. Math. Program. 162, 431–463 (2017). https://doi.org/10.1007/s10107-016-1049-8

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  • DOI: https://doi.org/10.1007/s10107-016-1049-8

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