The recent debate on hyper-computation has raised new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics. We propose here the idea of geometry of effective physical process as the essentially physical notion of computation. In Quantum mechanics we cannot use the traditional Euclidean geometry but we introduce more sophisticate non Euclidean geometry which include a new kind of information diffuse in the entire universe and that we can represent as Fisher information or active information. We remark that from the Fisher information we can obtain the Bohm and Hiley quantum potential and the classical Schrodinger equation. We can see the quantum phenomena do not affect a limited region of the space but is reflected in a change of the geometry of all the universe. In conclusion any local physical change or physical process is reflected in all the universe by the change of its geometry, This is the deepest meaning of the entanglement in Quantum mechanics and quantum computing. We stress the connection between metric and information as measure of change. Because computation is not restricted to calculus but is the environment changing via physical processes, super-Turing potentialities derive from an incomputable information source embedded into the geometry of the universe in accordance with Bell's constraints. In the general relativity we define the geometry of the space time. In our approach quantum phenomena define the geometry of the parameters of the probability distribution that include also the space time parameters. To study this new approach to the computation we use the new theory of Morphogenic systems.
The recent debate on hyper-computation has raised new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics. We propose here the idea of geometry of effective physical process as the essentially physical notion of computation. In Quantum mechanics we cannot use the traditional Euclidean geometry but we introduce more sophisticate non Euclidean geometry which include a new kind of information diffuse in the entire universe and that we can represent as Fisher information or active information. We remark that from the Fisher information we can obtain the Bohm and Hiley quantum potential and the classical Schrodinger equation. We can see the quantum phenomena do not affect a limited region of the space but is reflected in a change of the geometry of all the universe. In conclusion any local physical change or physical process is reflected in all the universe by the change of its geometry, This is the deepest meaning of the entanglement in Quantum mechanics and quantum computing. We stress the connection between metric and information as measure of change. Because computation is not restricted to calculus but is the environment changing via physical processes, super-Turing potentialities derive from an incomputable information source embedded into the geometry of the universe in accordance with Bell's constraints. In the general relativity we define the geometry of the space time. In our approach quantum phenomena define the geometry of the parameters of the probability distribution that include also the space time parameters. To study this new approach to the computation we use the new theory of Morphogenic systems.
The recent debate on hypercomputation has arisen new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics. We propose here the idea of "effective physical process" as the essentially physical notion of computation. By using the Bohm and Hiley active information concept we analyze the differences between the standard form (quantum gates) and the non-standard one (adiabatic and morphogenetic) of Quantum Computing, and we point out how its Super-Turing potentialities derive from an incomputable information source in accordance with Bell's constraints. On condition that we give up the formal concept of "universality", the possibility to realize quantum oracles is reachable. In this way computation is led back to the logic of physical world.
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.
Quantum information processing presents significant challenges due to the inherent fragility of quantum states. Classical geometric models often struggle to capture the complexities of quantum behavior, particularly concerning the effects of measurement and entanglement. This paper proposes a novel framework for representing and processing quantum information centered around non-Euclidean geometries, specifically emphasizing the principles of conformal invariance. We argue that this approach offers a fundamentally different perspective on quantum state manipulation, potentially enabling more robust and efficient computations. The core mechanism involves formalizing quantum state representations using a non-Euclidean metric space, allowing for a new computational paradigm that overcomes limitations inherent in traditional geometric approaches. This work addresses the limitations of classical geometry and lays the groundwork for a new era in quantum information processing.
The pursuit of a fundamentally new geometric paradigm challenges traditional notions of space and distance. This paper explores a novel approach to geometry based on quantum entanglement propagation, proposing a mechanism that allows for instantaneous information transfer across distances, potentially surpassing the constraints of classical spacetime. We demonstrate a crucial principle – that entangled quantum systems can effectively transmit geometric data, offering a pathway to a non-Euclidean geometry. The core of this investigation lies in a modified entanglement protocol that incorporates a dynamic 'signature' that guides the propagation of geometric information. The implications of this research extend beyond theoretical novelty, suggesting a potential revolution in computation and understanding of spacetime.
This paper introduces the concept of Quantum Information Curvature (QIC), proposing that information space possesses a curvature that fundamentally influences computation and physical processes. The curvature, denoted as [R_I], is hypothesized to arise from the complex interactions inherent in quantum systems and the constraints imposed by information theory. We explore the potential implications of this curvature, suggesting a novel framework for understanding quantum phenomena by connecting information, geometry, and physics. The core argument centers on the idea that deviations from Euclidean information space, manifesting as curvature, are not merely mathematical artifacts but represent a tangible physical effect. This framework opens avenues for exploring new computational paradigms and potentially resolving longstanding challenges in quantum measurement and decoherence. Further research is needed to rigorously define and quantify the curvature, but this initial conceptualization provides a foundational step towards a more holistic understanding of quantum information. ---
According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Several idealised computational models are known which suggest the possibility of hypercomputation, some Newtonian, some based on cosmology, some on quantum theory. While these models' physicality is debatable, they nonetheless throw into question the validity of extending CTT to include all physical systems. We consider the physicality of hypercomputational behaviour from first principles, by showing that quantum theory can be reformulated in a way that explains why physical behaviours can be regarded as 'computing something' in the standard computational state-machine sense. While this does not rule out the physicality of hypercomputation, it strongly limits the forms it can take. Our model also has physical consequences; in particular, the continuity of motion and arrow of time become theorems within the basic model.