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Vertex structure of fiber products of probability polytopes

Aziz Kharoof, Cihan Okay

Abstract

We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.

Vertex structure of fiber products of probability polytopes

Abstract

We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.
Paper Structure (16 sections, 38 theorems, 232 equations, 9 figures)

This paper contains 16 sections, 38 theorems, 232 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Vertices of polytopes in standard form
  4. Polytopes in standard form
  5. Vertices of fiber products of polytopes
  6. Extremal simplicial distributions
  7. Simplicial distributions
  8. Gluing measurement spaces
  9. Distributions on graphs
  10. Complete bipartite graph
  11. Rose graph
  12. Dipole graph
  13. Graph-theoretic characterization of vertices
  14. Counting the number of vertices
  15. Collapsing to the rose graph
  16. ...and 1 more sections

Key Result

Theorem 1

Let $x=(x^{(1)},\dots,x^{(n)})$ be a point in the fiber product $L^{(1)} \times_M \cdots \times_M L^{(n)}$. If, for every $1 \leq i \leq n$, the map $f_i$ is injective on $\operatorname{Vsupp}(x^{(i)})$ and the set $A_i$ is affinely independent, and if the intersection $\bigcap_{i=1}^n \hbox{\sf Con

Figures (9)

  • Figure 1: Basic graph families: the complete bipartite graph $K_{3,3}$, the rose graph $R_4$, and the dipole graph $D_5$.
  • Figure 2: The complete bipartite graph $K_{3,3}$.
  • Figure 3: The rose graph $R_4$, obtained by gluing four circles at a single vertex.
  • Figure 4: The dipole graph $D_5$.
  • Figure 5: The bipartite graphs $H(A_1)$, $H(A_2)$, $H(A_3)$, $H(A_4)$ from Example \ref{['ex:simplistforwedgetwopoints']}, with connected components indicated by different colors.
  • ...and 4 more figures

Theorems & Definitions (125)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 115 more