Revisiting Born's rule through Uhlhorn's and Gleason's theorems

TL;DR

This paper revisits Born's rule by connecting it to Gleason's and Uhlhorn's theorems, proposing a derivation rooted in physical axioms that clarifies the role of unitary transformations in quantum mechanics.

Contribution

It extends previous work by demonstrating that the necessity of unitary transforms between contexts follows from Uhlhorn's theorem, removing the need for this assumption.

Findings

Born's rule can be derived from physical axioms without assuming unitary transforms.

Uhlhorn's theorem explains the necessity of unitary transformations between quantum contexts.

The approach strengthens the link between quantum foundations and mathematical theorems.

Abstract

In a previous article [1] we presented an argument to obtain (or rather infer) Born's rule, based on a simple set of axioms named "Contexts, Systems and Modalities" (CSM). In this approach there is no "emergence", but the structure of quantum mechanics can be attributed to an interplay between the quantized number of modalities that are accessible to a quantum system, and the continuum of contexts that are required to define these modalities. The strong link of this derivation with Gleason's theorem was emphasized, with the argument that CSM provides a physical justification for Gleason's hypotheses. Here we extend this result by showing that an essential one among these hypotheses - the need of unitary transforms to relate different contexts - can be removed and is better seen as a necessary consequence of Uhlhorn's theorem.