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Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms

Sébastien Designolle, Tamás Vértesi, Sebastian Pokutta

Abstract

In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms, and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrised local polytope and they give the best known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalise our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.

Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms

Abstract

In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms, and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrised local polytope and they give the best known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalise our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.
Paper Structure (5 sections, 29 equations, 1 figure, 1 table)

This paper contains 5 sections, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simple example
  4. Group action
  5. Symmetrised local polytopes

Figures (1)

  • Figure 1: The symmetrised local polytope for $N=2$ and $m=3$, that is, its projection on the subspace of matrices of the form of \ref{['eqn:subspace_example']}. The point $\mathbf{r}^{(3)}_2$ outside of the polytope is given in \ref{['eqn:GHZmatrix']}.