Local Polytope Overview

Compute the bounds of classical Bell scenarios.

The LocalPolytope.jl module provides tools for characterizing the convex polyhedral structure of the correlations attainable using classical resources in a Bell scenario.

Each distinct strategy for a given Bell scenario corresponds to a point in vector space. The complete set of attainable strategies for a particular Bell scenario is denoted $\mathbf{P}$. Given the constraints of normalization and non-negativity, the extreme points of $\mathbf{P}$ correspond to deterministic strategies (matrices with 0,1 elements). Furthermore, when shared randomness is used in a Bell scenario, any two strategies can be mixed together in a convex combination. Hence, $\mathbf{P}$ is a convex polyhedron referred to as the local polytope.

A convex polytope has two equivalent descriptions:

1. V-Description: The polytope is the convex hull of a set of extreme points known as vertices $\mathbf{V}$, $\quad\mathbf{P} = \text{conv}(\mathbf{V})$.
2. H-Description: The polytope is the intersection of a set of linear half-spaces known as facets $\mathbf{F}$, $\quad\mathbf{P} = \cap\mathbf{F}$.

These two descriptions are equivalent,

\[\mathbf{P} = \text{conv}(\mathbf{V}) = \cap\mathbf{F},\]

and there exists a transformation between the set of vertices and facets $\mathbf{V} \leftrightarrow \mathbf{F}$.

For a given Bell scenario, the V-Description is typically easier to compute, however, the H-Description describes the bounds of the local polytope as as linear inequalities known as Bell inequalities. Bell inequalities are important because they provide a simple test to verify that a strategy is not contained by the local polyotpe. That is, if a Bell inequality is violated, the classical resources considered for the local polytope are not sufficient to reproduce the strategy. Hence, a Bell violation witnesses the use of resources with greater operational value. Bell violations are often used to characterize the advantages of quantum resources, however, it is important to note that a Bell violation can simply mean that more classical resources were used than anticipated.

Module Exports:

• vertices - Compute the set of extreme-points for the local polytope.
• num_vertices - The number of vertices for the local polytope.
• vrep - Construct a Polyhedron in the vertex representation.
• facets - Compute the linear inequalities which bound the local polytope.
• generator_vertex - Provide a canonical form for a vertex.
• generator_facet - Provide a canonical form for a facet.
• adjacency_decomposition - Efficiently compute the generator facets for the local polytope using the adjacency decomposition technique.