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Improved Local Models and New Bell Inequalities Via Frank-Wolfe Algorithms (2023)

Designolle, Sébastien ; Besançon, Mathieu ; Iommazzo, Gabriele ; [et al.]

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Publication Date: 2024-01-24

Description: In Bell scenarios with two outcomes per party, we algorithmically consider the two sides of the membership problem for the local polytope: Constructing local models and deriving separating hyperplanes, that is, Bell inequalities. We take advantage of the recent developments in so-called Frank-Wolfe algorithms to significantly increase the convergence rate of existing methods. First, we study the threshold value for the nonlocality of two-qubit Werner states under projective measurements. Here, we improve on both the upper and lower bounds present in the literature. Importantly, our bounds are entirely analytical; moreover, they yield refined bounds on the value of the Grothendieck constant of order three: 1.4367⩽KG(3)⩽1.4546. Second, we demonstrate the efficiency of our approach in multipartite Bell scenarios, and present local models for all projective measurements with visibilities noticeably higher than the entanglement threshold. We make our entire code accessible as a julia library called BellPolytopes.jl.

Language: English

Type: article , doc-type:article

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Cycle-based formulations in distance geometry (2023)

Liberti, Leo ; Iommazzo, Gabriele ; Lavor, Carlile ; [et al.]

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Publication Date: 2024-01-31

Description: The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension , where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, the formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.

Language: English

Type: article , doc-type:article

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Improved local models and new Bell inequalities via Frank-Wolfe algorithms (2023)

Designolle, Sébastien ; Iommazzo, Gabriele ; Besançon, Mathieu ; [et al.]

add to mindlist on the mindlist

Details

Publication Date: 2024-01-31

Description: In Bell scenarios with two outcomes per party, we algorithmically consider the two sides of the membership problem for the local polytope: Constructing local models and deriving separating hyperplanes, that is, Bell inequalities. We take advantage of the recent developments in so-called Frank-Wolfe algorithms to significantly increase the convergence rate of existing methods. First, we study the threshold value for the nonlocality of two-qubit Werner states under projective measurements. Here, we improve on both the upper and lower bounds present in the literature. Importantly, our bounds are entirely analytical; moreover, they yield refined bounds on the value of the Grothendieck constant of order three: 1.4367⩽KG(3)⩽1.4546. Second, we demonstrate the efficiency of our approach in multipartite Bell scenarios, and present local models for all projective measurements with visibilities noticeably higher than the entanglement threshold. We make our entire code accessible as a julia library called BellPolytopes.jl.

Language: English

Type: article , doc-type:article

Permalink