Symmetry and Self-Duality in Categories of Probabilistic Models

TL;DR

This paper presents axiomatic frameworks that derive the Jordan algebraic structure of finite-dimensional quantum theory using symmetry and self-duality properties, leveraging the Koecher-Vinberg Theorem.

Contribution

It introduces two axiomatic packages that facilitate deriving quantum structures from symmetry and self-duality principles, expanding the foundational understanding of quantum theory.

Findings

Derivation of quantum structures from symmetry principles

Use of Koecher-Vinberg Theorem to connect order-unit spaces and Jordan algebras

Establishment of axiomatic frameworks for finite-dimensional quantum theory

Abstract

This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic structure of finite-dimensional quantum theory. The derivation relies on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit spaces having homogeneous, self-dual cones, and formally real Jordan algebras.