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arXiv:1005.4190

Loops, Loop Sequences and Loop Surfaces in Statistical Geodesics, and Quantum Information

Aalok Pandya

Published May 23, 2010• 0 citations
arXivHTMLPDF

Abstract

The locus of probability flow in Quantum Mechanics and information is explored. We explore loops, loop sequences and loop surfaces in the statistical geodesics. Having known about the loop character of the statistical geodesics in probability space, we discuss wider physical aspects of consequences of the geometry of the probabilities in the form of loop sequences and loop surfaces. These are interpreted as the locus of probability flow. A deeper philosophical observation reveals that these loops, loop sequences and loop surfaces are the information grids. These investigations in many ways could be significant not only in the research in Statistics, but also in Mathematics, and Physics in general, and entropy, gravity and information theory in specific.

DOI
10.48550/arxiv.1005.4190

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The locus of probability flow in Quantum Mechanics and information is explored. We explore loops, loop sequences and loop surfaces in the statistical geodesics. Having known about the loop character of the statistical geodesics in probability space, we discuss wider physical aspects of consequences of the geometry of the probabilities in the form of loop sequences and loop surfaces. These are interpreted as the locus of probability flow. A deeper philosophical observation reveals that these loops, loop sequences and loop surfaces are the information grids. These investigations in many ways could be significant not only in the research in Statistics, but also in Mathematics, and Physics in general, and entropy, gravity and information theory in specific.
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Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
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We present a probabilistic-geometric formulation of quantum motion that establishes a direct geometric correspondence between statistical uncertainty and relativistic curvature. Quantum states are represented as probability distributions on a statistical manifold equipped with the Fisher–Rao metric, whose geodesic flow reproduces the dynamical evolution of quantum motion in probabilistic form. Embedding this manifold into a Lorentzian spacetime yields a unified metric structure in which stochastic fluctuations generate microscopic deformations of spacetime geometry. This framework enables the visualization and prediction of quantum motion as probabilistic geodesics, and provides a foundation for connecting information geometry, stochastic mechanics, and general relativity within a single coherent formulation.
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