A diagrammatic formulation of local realism

TL;DR

This paper introduces a diagrammatic approach to local realism in quantum mechanics, showing how locality and realism are represented by a single diagram and deriving the CHSH inequality from this formulation.

Contribution

It provides a novel diagrammatic formulation of local realism using category theory, linking it to the CHSH inequality and emphasizing the role of non-commutativity in quantum theory.

Findings

Diagrammatic formulation captures local realism principles.

Derives CHSH inequality from the diagrammatic framework.

Highlights the role of non-commutativity in quantum mechanics.

Abstract

Given two parties performing experiments in separate laboratories, we provide a diagrammatic formulation of what it means for the joint statistics of their experiments to satisfy local realism. In particular, we show that the principles of locality and realism are both captured by a single commutative diagram in the category of probability-preserving maps between finite probability spaces, and we also show that an assumption of such a diagrammatic formulation of local realism implies the standard CHSH inequality associated with dichotomic random variables. As quantum theory is known not to satisfy local realism, our formulation of local realism in terms of commutative diagrams provides yet another way in which the notion of non-commutativity plays a fundamental role in quantum theory. We note that we do not assume any prior knowledge of category theory or quantum theory, as this work is intended for philosophers, mathematicians and physicists alike.