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Bell’s Nonlocality in a General Nonsignaling Case: Quantitatively and Conceptually

Abstract

Quantum violation of Bell inequalities is now used in many quantum information applications and it is important to analyze it both quantitatively and conceptually. In the present paper, we analyze violation of multipartite Bell inequalities via the local probability model—the LqHV (local quasi hidden variable) model (Loubenets in J Math Phys 53:022201, 2012), incorporating the LHV model only as a particular case and correctly reproducing the probabilistic description of every quantum correlation scenario, more generally, every nonsignaling scenario. The LqHV probability framework allows us to construct nonsignaling analogs of Bell inequalities and to specify parameters quantifying violation of Bell inequalities—Bell’s nonlocality—in a general nonsignaling case. For quantum correlation scenarios on an N-qudit state, we evaluate these nonlocality parameters analytically in terms of dilation characteristics of an N-qudit state and also, numerically—in d and N. In view of our rigorous mathematical description of Bell’s nonlocality in a general nonsignaling case via the local probability model, we argue that violation of Bell inequalities in a quantum case is not due to violation of the Einstein–Podolsky–Rosen (EPR) locality conjectured by Bell but due to the improper HV modelling of “quantum realism”.

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Notes

  1. See Introductions in [4,5,6,7,8] and discussions in [9,10,11,12,13,14].

  2. For the general framework on Bell inequalities, see [15].

  3. That is, Bell inequalities of an arbitrary type—either on correlation functions or on joint probabilities or of a more complicated form.

  4. On this notion, see section 3 in [5], also, Sect. 2 below.

  5. For the general framework on the probabilistic description of multipartite correlation scenarios, see [5].

  6. For the main statements on the LHV modelling of a general multipartite correlation scenario, see section 4 in [5].

  7. This terminology formed historically, see Introduction in [5].

  8. For this decomposition see, for example, section 3 in [6].

  9. For these Bell inequalities, see, for example, subsection 3.2 in [15].

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Acknowledgements

The valuable discussions with Professor A. Khrennikov are very much appreciated.

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Correspondence to Elena R. Loubenets.

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Loubenets, E.R. Bell’s Nonlocality in a General Nonsignaling Case: Quantitatively and Conceptually. Found Phys 47, 1100–1114 (2017). https://doi.org/10.1007/s10701-017-0077-4

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Keywords

  • Nonsignaling
  • Bell’s nonlocality
  • The LqHV modelling
  • Quantum realism