The Born rule as structure of spectral bundles (extended abstract)

TL;DR

This paper presents a unified topos-theoretic framework using spectral bundles to understand quantum foundations, with a focus on the Born rule's probabilistic predictions as sections of valuation bundles.

Contribution

It introduces spectral bundles as a unified approach to quantum foundations and models the Born rule probabilistic predictions within this geometric bundle framework.

Findings

Spectral bundles encode quantum contextuality.

Born rule predictions are modeled as sections of valuation bundles.

The geometric valuation locale monad is central to the construction.

Abstract

Topos approaches to quantum foundations are described in a unified way by means of spectral bundles, where the base space is a space of contexts and each fibre is its spectrum. Differences in variance are due to the bundle being a fibration or opfibration. Relative to this structure, the probabilistic predictions of the Born rule in finite dimensional settings are then described as a section of a bundle of valuations. The construction uses in an essential way the geometric nature of the valuation locale monad.