2017/03/20 by Matthew A. Graydon, Graydon, Matthew A. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Categorical quantum mechanics #Category theory #Euclidean geometry #FOS: Physical sciences #Geometry #Hilbert space #Mathematical Physics (math-ph) #Mathematics #Morphism #Open quantum system #POVM #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum mechanics #Quantum operation #Quantum state #Theoretical physics #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1703.06800
PhD Thesis. 1+195 pages, 10pt, single line spacing
arxiv created 2017/03/20 · openalex publication_date 2017/03/20 · arxiv updated 2017/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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-Jamiołkowski 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.