This tutorial documents our opinion that quantum concepts are declarative in nature, and that using imperative programming for quantum computation is a conceptual mismatch. We show how to construct a purely functional framework that describes a methodologically more correct (in our opinion) model of quantum entities. We refrain from representing explicitly, as data structures, such objects as states which cannot be analysed without being destroyed. We suggest that a declarative, purely functional approach gives formal tools that are more adequate for reasoning about the computational sense/contents of quantum entities, and in general of several mathematical constructs considered abstract. We show that this approach can be used in practice, and we suggest that it also can shed some light on the sense of the linearity -- the geometric property of vector spaces -- within the quantum framework.
This tutorial documents our opinion that quantum concepts are declarative in nature, and that using imperative programming for quantum computation is a conceptual mismatch. We show how to construct a purely functional framework that describes a methodologically more correct (in our opinion) model of quantum entities. We refrain from representing explicitly, as data structures, such objects as states which cannot be analysed without being destroyed. We suggest that a declarative, purely functional approach gives formal tools that are more adequate for reasoning about the computational sense/contents of quantum entities, and in general of several mathematical constructs considered abstract. We show that this approach can be used in practice, and we suggest that it also can shed some light on the sense of the linearity -- the geometric property of vector spaces -- within the quantum framework.
We present a framework for representing quantum entities in Haskell. States and operators are functional objects, and their semantics is defined --- as far as possible --- independently of the base in the Hilbert space. We construct effectively the tensor states for composed systems, and we present a toy model of quantum circuit toolbox. We conclude that functional languages are right tools for formal computations in quantum physics. The paper focuses mainly on the representation, not on computational problems.
Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between general linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
An examination of the postulates of quantum mechanics reveals a number of fundamental mathematical constructs that form its theoretical underpinnings. Many of the results that are summarized in this Chapter will only become clear after reading the rest of the book and a re-reading may be in order. The dynamical variables of classical mechanics are superseded by the quantum degree of freedom. An exhaustive and complete description of the indeterminate degree of freedom is given by its state function, which is an element of a state space that, in general, is an infinite-dimensional linear vector space. The properties of the indeterminate degree of freedom are extracted from its state vector by the linear action of operators representing experimentally observable quantities. Repeated applications of the operators on the state function yield the average value of the operator for the state [Baaquie (2013e)]. The conceptual framework of quantum mechanics is discussed in Section 2.1. The concepts of degree of freedom, state space and operators are briefly reviewed in Sections 2.3–2.5. Three distinct formulations of quantum mechanics emerge from the superstructure of quantum mechanics and these are briefly summarized in Sections 2.7–2.9. The Copenhagen quantum postulate The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohr and Werner Heisenberg, provides a conceptual framework for the interpretation of the mathematical constructs of quantum mechanics and is the standard interpretation that is followed by the majority of practicing physicists [Stapp (1963), Dirac (1999)].
This paper explores the potential of non-Euclidean geometry to provide a novel framework for understanding and modeling quantum information. Traditional approaches often treat quantum information as a purely mathematical construct, limiting its applicability to purely computational contexts. We propose a radical shift, formalizing quantum information as a geometric object within a non-Euclidean space, leveraging the properties of geometry to reveal deeper connections to the physical world. The core mechanism centers around extending the concept of quantum information beyond classical representations, allowing for the visualization and manipulation of entanglement and other quantum phenomena through a geometric lens. This paper will detail the conceptual groundwork, mathematical framework, and potential implications of this approach, emphasizing its capacity to bridge the gap between abstract mathematics and the complexities of quantum mechanics. Chapter 1: Foundations – Geometry and Information The fundamental concept underpinning this research lies in the recognition that geometry, particularly non-Euclidean geometry, possesses inherent properties that resonate with the nature of quantum information. Classical geometry, with its inherent Euclidean assumptions, struggles to adequately capture the counterintuitive aspects of quantum mechanics. The inherent non-Euclidean nature of space, where distances are not absolute but relative, mirrors the fundamental fuzziness and probabilistic nature of quantum states. Furthermore, the distribution of information, a cornerstone of quantum mechanics, can be visualized as a complex geometric pattern, and non-Euclidean geometry offers a powerful tool to analyze and manipulate this pattern. The challenge, therefore, is to move beyond a purely mathematical abstraction to a geometric understanding of information itself. Early attempts to model quantum information often relied on representations like Hilbert spaces, which, while powerful, represent information as a vector space. However, this representation lacks the inherent geometric richness that is crucial for understanding entanglement and other quantum phenomena. We propose a shift towards representing quantum information as points within a non-Euclidean space, where the distance and orientation of these points encode the quantum state. This shift is predicated on the observation that geometric properties, such as curvature and connectivity, are intimately linked to quantum entanglement and the preservation of information. The core idea is to formulate a geometric model where entanglement is represented not as a correlation between particles but as a distortion of the geometric space itself, influenced by the quantum state. This distortion, driven by the inherent properties of the non-Euclidean space, provides a richer and potentially more intuitive understanding of quantum phenomena. The choice of non-Euclidean geometry is not arbitrary; it is intended to be a 'natural' choice, reflecting the underlying structure of quantum information. Chapter 2: Non-Euclidean Geometry – A Conceptual Framework We introduce a specific type of non-Euclidean geometry, termed "Hyperbolic Geometry for Quantum Information," that incorporates several key features. Unlike Euclidean geometry, Hyperbolic Geometry allows for the representation of multiple, potentially conflicting, states simultaneously. This is achieved through the concept of "Hyperbolic Density," which quantifies the probability of finding a quantum state within a given region of the space. The density of a Hyperbolic Density is influenced by the quantum state, creating a dynamic and evolving geometric representation. The mathematical framework of Hyperbolic Geometry is built around the concept of "Geometric Curvature." This curvature is not simply a measure of distance but rather a representation of the quantum state's inherent orientation. Specifically, we define a curvature function that is sensitive to the quantum state. This curvature is not constant; it adapts to the quantum state, creating a continuously evolving geometric representation. This adaptation is crucial for capturing the inherent randomness and non-determinism of quantum mechanics. Furthermore, we introduce the concept of "Geometric Connectivity," which describes the relationships between points within the Hyperbolic Geometry. These connections are not simply spatial relationships but are defined by the quantum state's inherent properties. The strength and type of connectivity represent the degree to which the quantum state is entangled with the geometric structure. This connectivity is dynamically modified by the curvature function, creating a complex and interconnected geometric pattern. The core mathematical tool for manipulating these geometries is the "Geometric Transformation." This transformation allows us to 'morph' geometric forms into new forms, representing the evolution of quantum information. The transformation is based on a specific set of rules which are intrinsically linked to the nature of entanglement. These rules are carefully designed to preserve the fundamental properties of quantum information.
This component documents the development of the representation-theoretic framework of the Invariant Core Theory (IKT), designated as IKT-R. IKT-R extends the formal core of IKT by clarifying under which structural conditions common forms of physical representation become admissible. It analyzes time parameters, dynamical equations, Hilbert space structures, and geometric spacetime models as regime-dependent representational constructs derived from structural continuation constraints. Elements of this framework are summarized in condensed form in the canonical manuscript “Formal Structure and Consequences of the Invariant Core Theory” (P2; listet in the parent project as C2.1). The present document provides the extended representation-theoretic development The document reflects a stable research state reached through an extended period of structured theoretical development. Its formulation emerged in sustained conceptual collaboration between Ulrich Grob and the AI system “Nathan” (ChatGPT), used as a high-level structural reasoning partner under explicit methodological safeguards. This collaboration did not replace authorship or conceptual responsibility. All structural claims and formal conclusions were critically evaluated, revised, and consolidated within the documented research process. IKT-R is published here as an immutable archival record. It represents a foundational component of the broader IKT framework and provides the representation-theoretic basis for subsequent regime analyses (e.g., General Relativity, quantum mechanical representation, and cosmological modeling). The document is written in German. The structural content and formal arguments are independent of language.
Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables supplies the syntax; (b) states, viewed as linear functionals, give the semantics; and (c) a novel diagrammatic calculus unifies the two. The first part develops the basic objects of the framework, encoding consistent patterns of operator correlation, recovering Hilbert space via the GNS construction, and re-deriving the Bloch sphere as the set of all consistent correlations of operators in the Pauli algebra. We then turn to intervention, showing how measurement modifies state, proving an operator-algebraic version of the Knill-Laflamme conditions, and expressing stabilizer codes with the same diagrammatic machinery. This provides a concise, representation-agnostic account of quantum error correction. The result is a self-contained foundation in which C*-algebras, and their dual Hilbert spaces, offer a rich and universal substrate for quantum programming; forthcoming papers will build a high-level language and quantum software applications on top of this substrate.