Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories

TL;DR

This paper explores the geometric and categorical structures underlying quantum theory and its generalizations using conical designs and Jordan algebraic frameworks, aiming to deepen foundational understanding.

Contribution

It introduces conical designs and investigates categorical Jordan algebraic structures as post-quantum theories, expanding the geometric and algebraic landscape beyond standard quantum mechanics.

Findings

Characterization of quantum state spaces via conical designs

Categorical framework for Jordan algebraic post-quantum theories

Insights into the geometry of quantum and post-quantum state spaces

Abstract

Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamio{\l}kowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras. In posting this thesis on the arXiv, we hope that it might serve as a useful resource for those interested in its subjects.