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Extremal simplicial distributions on cycle scenarios with arbitrary outcomes

Aziz Kharoof, Cihan Okay, Selman Ipek

TL;DR

This work provides a complete classification of contextual vertices for cycle scenarios with arbitrary outcomes within the simplicial-distributions framework, showing that contextual vertices are exactly the $k$-order cycle distributions with $k\ge 2$ (and deterministic for $k=1$). The authors deploy homotopical methods and bundle perspectives to reduce vertex detection to tractable criteria on faces and to extend results to spaces formed by gluing cycles. They derive a closed-form count of vertices, $V_{n,d}=\sum_{k=1}^d \binom{d}{k}^n (k!)^{n-1} (k-1)!$, and demonstrate the theory on concrete examples including PR boxes, trichotomic-outcome scenarios, and the $(2,3,3)$ Bell case. The approach yields a principled route to vertex enumeration and has potential implications for measurement-based quantum computation and classical simulations of quantum phenomena, while inviting further exploration of gluing- and bundle-based generalizations.

Abstract

Cycle scenarios are a significant class of contextuality scenarios, with the Clauser-Horne-Shimony-Holt (CHSH) scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.

Extremal simplicial distributions on cycle scenarios with arbitrary outcomes

TL;DR

This work provides a complete classification of contextual vertices for cycle scenarios with arbitrary outcomes within the simplicial-distributions framework, showing that contextual vertices are exactly the -order cycle distributions with (and deterministic for ). The authors deploy homotopical methods and bundle perspectives to reduce vertex detection to tractable criteria on faces and to extend results to spaces formed by gluing cycles. They derive a closed-form count of vertices, , and demonstrate the theory on concrete examples including PR boxes, trichotomic-outcome scenarios, and the Bell case. The approach yields a principled route to vertex enumeration and has potential implications for measurement-based quantum computation and classical simulations of quantum phenomena, while inviting further exploration of gluing- and bundle-based generalizations.

Abstract

Cycle scenarios are a significant class of contextuality scenarios, with the Clauser-Horne-Shimony-Holt (CHSH) scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.
Paper Structure (16 sections, 17 theorems, 58 equations, 5 figures)

This paper contains 16 sections, 17 theorems, 58 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Homotopical vertices of faces
  4. Simplicial distributions
  5. The vertex support
  6. Null-homotopy
  7. Vertices of faces
  8. Bundle scenarios
  9. Simplicial distributions on bundles
  10. Morphisms of bundle scenarios
  11. Cycle scenario with arbitrary outcome
  12. Scenarios obtained by gluing
  13. Popescu-Rohrlich boxes
  14. A scenario with trichotomic measurements
  15. $(2,3,3)$ Bell scenario
  16. ...and 1 more sections

Key Result

Theorem 1

A simplicial distribution on the cycle scenario $C^{(n)}$ is a contextual vertex if and only if it is a $k$-order cycle distribution for some $k\geq 2$.

Figures (5)

  • Figure 1: The $4$-circle measurement space.
  • Figure 2: Decomposition of the $4$-circle.
  • Figure 3: The measurement space decomposed into two parts given by $2$-circles.
  • Figure 4:
  • Figure 5: (a) Space obtained from $K_{3,3}$ by collapsing the edges with distribution $T$ to a point. (b) One of the two copies of the edge with distribution $R$ is removed. (c) Measurement space decomposed into two parts.

Theorems & Definitions (43)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • proof
  • ...and 33 more