Local Polytope Overview
BellScenario.LocalPolytope
— ModuleCompute 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:
- V-Description: The polytope is the convex hull of a set of extreme points known as vertices $\mathbf{V}$, $\quad\mathbf{P} = \text{conv}(\mathbf{V})$.
- 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 aPolyhedron
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.