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https://github.com/zib-iol/bellpolytopes.jl
This julia package addresses the membership problem for local polytopes: it constructs Bell inequalities and local models in multipartite Bell scenarios with binary outcomes.
https://github.com/zib-iol/bellpolytopes.jl
bell-inequalities conditional-gradients frank-wolfe local-models local-polytope
Last synced: about 1 month ago
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This julia package addresses the membership problem for local polytopes: it constructs Bell inequalities and local models in multipartite Bell scenarios with binary outcomes.
- Host: GitHub
- URL: https://github.com/zib-iol/bellpolytopes.jl
- Owner: ZIB-IOL
- License: mit
- Created: 2023-02-08T13:17:23.000Z (almost 2 years ago)
- Default Branch: main
- Last Pushed: 2024-12-07T14:50:24.000Z (about 2 months ago)
- Last Synced: 2024-12-07T15:25:44.852Z (about 2 months ago)
- Topics: bell-inequalities, conditional-gradients, frank-wolfe, local-models, local-polytope
- Language: Julia
- Homepage:
- Size: 4.41 MB
- Stars: 3
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Citation: CITATION.bib
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README
# BellPolytopes.jl
[](https://zib-iol.github.io/BellPolytopes.jl/dev/)
[](https://github.com/zib-iol/BellPolytopes.jl/actions/workflows/CI.yml?query=branch%3Amain)
This package addresses the membership problem for local polytopes: it constructs Bell inequalities and local models in multipartite Bell scenarios with arbitrary settings.
The original article for which it was written can be found here:
> [Improved local models and new Bell inequalities via Frank-Wolfe algorithms](http://arxiv.org/abs/2302.04721).
## Installation
The most recent release is available via the julia package manager, e.g., with
```julia
using Pkg
Pkg.add("BellPolytopes")
```
or the main branch:
```julia
Pkg.add(url="https://github.com/ZIB-IOL/BellPolytopes.jl", rev="main")
```
## Getting started
Let's say we want to characterise the nonlocality threshold obtained with the two-qubit maximally entangled state and measurements whose Bloch vectors form an icosahedron.
Using `BellPolytopes.jl`, here is what the code looks like.
```julia
julia> using BellPolytopes, LinearAlgebra
julia> N = 2; # bipartite scenario
julia> rho = rho_GHZ(N) # two-qubit maximally entangled state
4×4 Matrix{Float64}:
0.5 0.0 0.0 0.5
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.5 0.0 0.0 0.5
julia> measurements_vec = icosahedron_vec() # Bloch vectors forming an icosahedron
6×3 Matrix{Float64}:
0.0 0.525731 0.850651
0.0 0.525731 -0.850651
0.525731 0.850651 0.0
0.525731 -0.850651 0.0
0.850651 0.0 0.525731
0.850651 0.0 -0.525731
julia> _, lower_bound, upper_bound, local_model, bell_inequality, _ =
nonlocality_threshold(measurements_vec, N; rho = rho);
julia> println([lower_bound, upper_bound])
[0.7784, 0.7784]
julia> p = correlation_tensor(measurements_vec, N; rho = rho)
6×6 Matrix{Float64}:
0.447214 -1.0 -0.447214 0.447214 0.447214 -0.447214
-1.0 0.447214 -0.447214 0.447214 -0.447214 0.447214
-0.447214 -0.447214 -0.447214 1.0 0.447214 0.447214
0.447214 0.447214 1.0 -0.447214 0.447214 0.447214
0.447214 -0.447214 0.447214 0.447214 1.0 0.447214
-0.447214 0.447214 0.447214 0.447214 0.447214 1.0
julia> final_iterate = sum(local_model.weights[i] * local_model.atoms[i] for i in 1:length(local_model));
julia> norm(final_iterate - lower_bound * p) < 1e-3 # checking local model
true
julia> local_bound(bell_inequality)[1] / dot(bell_inequality, p) # checking the Bell inequality
0.7783914488195466
```
## Under the hood
The computation is based on an efficient variant of the Frank-Wolfe algorithm to iteratively find the local point closest to the input correlation tensor.
See this recent [review](https://arxiv.org/abs/2211.14103) for an introduction to the method and the package [FrankWolfe.jl](https://github.com/ZIB-IOL/FrankWolfe.jl) for the implementation on which this package relies.
In a nutshell, each step gets closer to the objective point:
* either by moving towards a *good* vertex of the local polytope,
* or by astutely combining the vertices (or atoms) already found and stored in the *active set*.
```julia
julia> res = bell_frank_wolfe(p; v0=0.8, verbose=3, callback_interval=10^2, mode_last=-1);
Visibility: 0.8
Symmetric: true
#Inputs: 6
Dimension: 21
Intervals
Print: 100
Renorm: 100
Reduce: 10000
Upper: 1000
Increment: 1000
Iteration Primal Dual gap Time (sec) #It/sec #Atoms #LMO
100 8.7570e-03 6.0089e-02 6.9701e-04 1.4347e+05 11 26
200 5.9241e-03 5.4948e-02 9.4910e-04 2.1073e+05 16 33
300 3.5594e-03 3.4747e-02 1.1942e-03 2.5122e+05 18 40
400 1.9068e-03 3.4747e-02 1.3469e-03 2.9697e+05 16 42
500 1.8093e-03 5.7632e-06 1.5409e-03 3.2448e+05 14 48
Primal: 1.81e-03
Dual gap: 2.60e-08
Time: 1.63e-03
It/sec: 3.28e+05
#Atoms: 14
v_c ≤ 0.778392
```
## Going further
More examples can be found in the corresponding folder of the package.
They include the construction of a Bell inequality with a higher tolerance to noise as CHSH as well as multipartite instances.