Optimal non-signalling violations via tensor norms

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.

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    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.

  2. 2.

    Formally, the tensor M defines the inequality \(\langle M,P\rangle \le \omega _{\mathcal L}(M)\) for every \(P\in \mathcal L\).

  3. 3.

    Note that Lemma 5.7 applies on non-negative tensors, so we must use it on \(-R^-=|R^{-}|\).

  4. 4.

    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.

References

  1. 1.

    Acín, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)

    Article  Google Scholar 

  2. 2.

    Bavarian, M., Shor, P.W.: Information causality, Szemerédi-Trotter and algebraic variants of CHSH. In: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pp. 123–132 (2015)

  3. 3.

    Bell, J.S.: On the Einstein–Poldolsky–Rosen paradox. Physics 1, 195 (1964)

    Article  Google Scholar 

  4. 4.

    Bergh, J., Löfström, J.: Interpolation spaces: an introduction. Grundlehren der mathematischen Wissenschaften, vol. 223. Springer-Verlag, Berlin (1976)

    Google Scholar 

  5. 5.

    Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014)

    Article  Google Scholar 

  6. 6.

    Buhrman, H., Regev, O., Scarpa, G., de Wolf, R.: Near-optimal and explicit Bell inequality violations. Theory Comput. 8, 623–645 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)

    Google Scholar 

  8. 8.

    Degorre, J., Kaplan, M., Laplante, S., Roland, J.: The communication complexity of non-signalling distributions. In: International Symposium on Mathematical Foundations of Computer Science, pp. 270–281 (2009)

  9. 9.

    Feige, U., Lovasz, L.: Two-prover one-round proof systems: their power and their problem. In: Proceedings of the 24th ACM Symposium on Theory of Computing, pp. 733–741 (1992)

  10. 10.

    Goemans, M.: Chernoff bounds, and some applications 18.310 lecture notes. http://math.mit.edu/~goemans/18310S15/chernoff-notes.pdf (2015)

  11. 11.

    Ito, T.: Polynomial-space approximation of no-signaling provers. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 6198, pp. 140–151. Springer, Berlin (2010)

    Google Scholar 

  12. 12.

    Junge, M., Palazuelos, C.: Large violation of Bell inequalities with low entanglement. Commun. Math. Phys. 306, 695–746 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Junge, M., Palazuelos, C., Pérez-García, D., Villanueva, I., Wolf, M.M.: Unbounded violations of bipartite Bell inequalities via operator space theory. Commun. Math. Phys. 300, 715–739 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Junge, M., Palazuelos, C., Pérez-García, D., Villanueva, I., Wolf, M.M.: Operator space theory: a natural framework for Bell inequalities. Phys. Rev. Lett. 104, 170405 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lami, L., Palazuelos, C., Winter, A.: Ultimate data hiding in quantum mechanics and beyond. Commun. Math. Phys. 361(2), 661–708 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lancien, C., Winter, A.: Parallel repetition and concentration for (sub-)no-signalling games via a flexible constrained de Finetti reduction. Chicago J. Theor. Comput. Sci. 2016, 4 (2016)

  17. 17.

    Palazuelos, C., Vidick, T.: Survey on nonlocal games and operator space theory. J. Math. Phys. 57, 015220 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Tomczak-Jaegermann, N.: Banach–Mazur Distances and Finite Dimensional Operator Ideals. Longman Scientific & Technical, Harlow (1989)

    Google Scholar 

  19. 19.

    Tsirelson, B.S.: Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Sov. Math. 36(4), 557–570 (1987)

    Article  Google Scholar 

  20. 20.

    Tsirelson, B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8(4), 329–345 (1993)

    MathSciNet  MATH  Google Scholar 

Download references

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.).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Abderramán Amr Rey.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Quantum information
  • Bell inequalities
  • Non-signalling distributions
  • Tensor norms

Mathematics Subject Classification

  • 81P45
  • 46B28