Completely hereditarily atomic OMLs

An irreducible complete atomic OML of infinite height cannot both be algebraic and have the covering property. However, Kalmbach's construction provides an example of such an OML that is algebraic and has the 2-covering property, and Keller's construction provides an example of such an OML that has the covering property and is completely hereditarily atomic. Completely hereditarily atomic OMLs generalize algebraic OMLs suitably to quantum predicate logic.Comment: 24 page

An effect-theoretic reconstruction of quantum theory

An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C*-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory. In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect. We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C*-algebras, which completes our reconstruction.Comment: 20+5 pages. V4: Journal version V3: complete rewrite. Changed name of the paper from the original 'Reconstruction of quantum theory from universal filters' to reflect a change of presentatio

Quantum Theory from Principles, Quantum Software from Diagrams

This thesis consists of two parts. The first part is about how quantum theory can be recovered from first principles, while the second part is about the application of diagrammatic reasoning, specifically the ZX-calculus, to practical problems in quantum computing. The main results of the first part include a reconstruction of quantum theory from principles related to properties of sequential measurement and a reconstruction based on properties of pure maps and the mathematics of effectus theory. It also includes a detailed study of JBW-algebras, a type of infinite-dimensional Jordan algebra motivated by von Neumann algebras. In the second part we find a new model for measurement-based quantum computing, study how measurement patterns in the one-way model can be simplified and find a new algorithm for extracting a unitary circuit from such patterns. We use these results to develop a circuit optimisation strategy that leads to a new normal form for Clifford circuits and reductions in the T-count of Clifford+T circuits.Comment: PhD Thesis. Part A is 135 pages. Part B is 95 page

Analyzing and controlling large nanosystems with physics-trained neural networks

In dieser Arbeit wird untersucht, wie Neuronale Netze genutzt werden können, um die Auswertung von Experimenten durch Minimierung des Simulationsaufwandes beschleunigen zu können. Für die Rekonstruktion von Silber-Nanoclustern aus Einzelschuss-Weitwinkel-Streubildern können diese bereits aus kleinen Datenätzen allgemeine Rekonstruktionsregeln ableiten und ermöglichen durch direktes Training auf der Streuphysik unerreichte Detailtiefen. Für Giant-Dipole-Zustände von Rydbergexzitonen in Kupferoxydul wird mittels Deep Reinforcement Learning ein Anregungsschema aus Simulationen hergeleitet.This thesis investigates the possible application of neural networks in accelerating the evaluation of physical experiments while minimizing the required simulation effort. Neural networks are capable of inferring universal reconstruction rules for reconstructing silver nanoclusters from single wide-angle scattering patterns from a small set of simulated data and when trained directly on scattering theory reaching unmatched accuracy. A dynamic excitation for giant dipole states of Rydberg excitons in cuprous oxide is derived through deep reinforcement learning interacting and simulation data

Glosarium Matematika

273 p.; 24 cm

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout

Categorical Quantum Dynamics

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford University, Michaelmas Term 2016 (273 pages