Cohomology of Effect Algebras

TL;DR

This paper introduces two novel cohomology theories for effect algebras in quantum logic, linking algebraic structures to quantum state spaces and foundational no-go theorems.

Contribution

It develops two distinct cohomology frameworks for effect algebras, enhancing understanding of their structure and implications in quantum foundations.

Findings

Cohomology groups relate to the state space of effect algebras.

Variations of Kunneth and Mayer-Vietoris sequences enable computations.

Cohomology characterizes state extensions and supports no-go theorems.

Abstract

We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.