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A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…
The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical…
Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction…
Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in…
Passive documents and active programs now widely comingle. Document languages include Turing-complete programming elements, and programming languages include sophisticated document notations. However, there are no formal foundations that…
The application of diagrammatic reasoning techniques to large-scale quantum processes needs specific tools to describe families of diagrams of arbitrary size. For now, large-scale diagrammatic reasoning tools in ZH-calculus come in two…
Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general,…
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
In this thesis we present a semantic representation formalism based on directed graphs and explore its linguistic adequacy and explanatory benefits in the semantics of plurality and quantification. Our graph language covers the essentials…
Theories of natural language and concepts have been unable to model the flexibility, creativity, context-dependence, and emergence, exhibited by words, concepts and their combinations. The mathematical formalism of quantum theory has…
We extend the synthetic theories of discrete and Gaussian categorical probability by introducing a diagrammatic calculus for reasoning about hybrid probabilistic models in which continuous random variables, conditioned on discrete ones,…
Given two parties performing experiments in separate laboratories, we provide a diagrammatic formulation of what it means for the joint statistics of their experiments to satisfy local realism. In particular, we show that the principles of…
Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning…
This invited paper presents an overview of an ongoing research program aimed at extending the Curry-Howard-Lambek correspondence to quantum computation. We explore two key frameworks that provide both logical and computational foundations…
We introduce a class of rooted graphs which allows one to encode various kinds of classical or quantum circuits. We then follow a set-theoretic approach to define rewrite systems over the considered graphs and propose a new complete…
Quantum theory is a probabilistic theory with fixed causal structure. General relativity is a deterministic theory but where the causal structure is dynamic. It is reasonable to expect that quantum gravity will be a probabilistic theory…
We take quantum theory and replace $\mathbb{C}$ by $\mathbb{C}[\varepsilon]$ where $\varepsilon^2=0$, i.e. we extend quantum theory to the ring of dual complex numbers. The aim is to develop a common language in which to treat continuous…
Traditional treatments of formal logic provide: 1. A syntax for formulas. 2. An inference relation between sets of formulas. 3. A rule for assigning meaning to formulas (semantics) that is sound with respect to the inference relation. First…
Quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a…
This paper examines language modeling based on the theory of quantum mechanics. It focuses on the introduction of quantum mechanics into the symbol-meaning pairs of language in order to build a representation model of natural language. At…