The space of measurement outcomes as a spectrum for non-commutative algebras

TL;DR

This paper explores the structure of measurement outcomes in non-commutative algebras using topos theory, revealing a spectrum that generalizes classical notions for quantum systems.

Contribution

It introduces a novel perspective on measurement outcomes as a spectrum for non-commutative algebras via Bohrification and sheafification techniques.

Findings

The space of measurement outcomes aligns with the spectrum for commutative C*-algebras.

Bohrification provides a locale of hidden variables within a topos.

Double negation sheafification yields a generalized space of measurement outcomes.

Abstract

Bohrification defines a locale of hidden variables internal in a topos. We find that externally this is the space of partial measurement outcomes. By considering the double negation sheafification, we obtain the space of measurement outcomes which coincides with the spectrum for commutative C*-algebras.