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On Contextuality as a Feature of Logic and Probability Theory

Ask Ellingsen

Abstract

In quantum mechanics, not everything that can be observed can be observed simultaneously. Observational data exhibits \emph{contextuality} -- a generalisation of nonlocality -- if the result of an observation is necessarily dependent on which combination of observables was measured. This article gives a mathematical introduction to contextuality, emphasising its nature as a general feature of probability theory and logic, rather than of any particular quantum theory.

On Contextuality as a Feature of Logic and Probability Theory

Abstract

In quantum mechanics, not everything that can be observed can be observed simultaneously. Observational data exhibits \emph{contextuality} -- a generalisation of nonlocality -- if the result of an observation is necessarily dependent on which combination of observables was measured. This article gives a mathematical introduction to contextuality, emphasising its nature as a general feature of probability theory and logic, rather than of any particular quantum theory.
Paper Structure (12 sections, 29 equations, 1 figure, 1 table)

This paper contains 12 sections, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Alice and Bob Receive Letters
  3. Quantum Probability
  4. The Born rule
  5. Contexts in QM
  6. Logic, Probability, and Order
  7. What is probability?
  8. Measure theoretic probability
  9. Boolean algebras
  10. Partial Boolean algebras
  11. The Kochen-Specker Theorem
  12. Outlook

Figures (1)

  • Figure 1: Bundle diagrams corresponding to the tables in Table \ref{['tab:tables']}. Darker colours generally correspond to higher probabilities. Joint results that are assigned zero probability are shown as dotted lines.

Theorems & Definitions (3)

  • proof
  • proof
  • proof