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Twisted simplicial distributions

Cihan Okay, Walker H. Stern

Abstract

We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell's non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.

Twisted simplicial distributions

Abstract

We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell's non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.
Paper Structure (21 sections, 64 theorems, 154 equations)

This paper contains 21 sections, 64 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Simplicial principal bundles
  4. Preliminaries
  5. Simplicial principal bundles
  6. The theory for a simplicial Abelian group
  7. The case of an Abelian group
  8. The tensor product of principal bundles
  9. Twisted distributions
  10. The category of $K$-bundle scenarios
  11. The colimit decomposition
  12. The polytope of twisted distributions
  13. The convolution product
  14. Equivariant distributions
  15. Contextuality
  16. ...and 6 more sections

Key Result

Theorem 1

There is a convex bijection

Theorems & Definitions (150)

  • Theorem
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 140 more