# Gleason-type Theorems from Cauchy's Functional Equation

## TL;DR

This paper offers an alternative proof of Gleason-type theorems in quantum mechanics by leveraging solutions to Cauchy's functional equation, connecting probability additivity to functional equation techniques.

## Contribution

It introduces a novel proof method for Gleason-type theorems using functional equations, expanding the mathematical tools available for foundational quantum theory.

## Key findings

- Provides a new proof of Gleason-type theorems
- Links additivity in quantum probabilities to Cauchy's functional equation
- Enhances understanding of the mathematical structure underlying quantum measurement

## Abstract

Gleason-type theorems derive the density operator and the Born rule formalism of quantum theory from the measurement postulate, by considering additive functions which assign probabilities to measurement outcomes. Additivity is also the defining property of solutions to Cauchy's functional equation. This observation suggests an alternative proof of the strongest known Gleason-type theorem, based on techniques used to solve functional equations.