Abstract
In this paper we characterize the set of bipartite non-signalling probability distributions in terms of tensor norms. Using this characterization we give optimal upper and lower bounds on Bell inequality violations when non-signalling distributions are considered. Interestingly, our upper bounds show that non-signalling Bell inequality violations cannot be significantly larger than quantum Bell inequality violations.
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Notes
Observe that both quantities depend on N and K, so we should denote \(LV^{N,K}_{\mathcal Q}\) and \(LV^{N,K}_{{\mathcal {NS}}}\), but we will simplify notation when N and K are clear from the context.
Formally, the tensor M defines the inequality \(\langle M,P\rangle \le \omega _{\mathcal L}(M)\) for every \(P\in \mathcal L\).
Note that Lemma 5.7 applies on non-negative tensors, so we must use it on \(-R^-=|R^{-}|\).
Note that, as we mentioned in Remark 5.3, in general we cannot replace \(\Vert \alpha _1(R)\Vert _{NSG\otimes _\pi NSG}\) by \(\Vert \alpha _1(R)\Vert _{\ell _\infty ^N(\ell _1^K)\otimes _\pi \ell _\infty ^N(\ell _1^K)}\). However, for the particular elements of the form \(Q_1\otimes Q_2\), both norms coincide by Lemma 5.2.
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Acknowledgements
This research was funded by the Spanish MINECO through Grant No. MTM2017-88385-P, MTM2014-54240-P and by the Comunidad de Madrid through grant QUITEMAD-CM P2018/TCS4342. We also acknowledge funding from SEV-2015-0554-16-3 and “Ramón y Cajal program” RYC-2012-10449 (C. P.).
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Rey, A.A., Palazuelos, C. & Villanueva, I. Optimal non-signalling violations via tensor norms. Rev Mat Complut 33, 661–694 (2020). https://doi.org/10.1007/s13163-019-00329-8
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DOI: https://doi.org/10.1007/s13163-019-00329-8