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Quanta

On the Connection Between Quantum Probability and Geometry

Federico Holik

Published Jun 18, 2021• 2 citations
Publisher

Abstract

We discuss the mathematical structures that underlie quantum probabilities. More specifically, we explore possible connections between logic, geometry and probability theory. We propose an interpretation that generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories. We stress the relevance of developing a geometrical interpretation of quantum mechanics. Quanta 2021; 10: 1–14.

DOI
10.12743/quanta.v10i1.148

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We discuss the mathematical structures that underlie quantum probabilities. More specifically, we explore possible connections between logic, geometry and probability theory. We propose an interpretation that generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories. We stress the relevance of developing a geometrical interpretation of quantum mechanics. Quanta 2021; 10: 1–14.
This paper explores the application of probabilistic geometry to the theoretical framework of quantum computation, specifically focusing on the development of a quantum geometric quantization model. We propose a novel approach that integrates geometric structures and probabilistic logic to provide a deeper understanding of quantum systems and to guide the optimization of quantum algorithms. The core mechanism centers around representing quantum states as probability distributions, allowing for a more nuanced and robust analysis of quantum phenomena. This framework offers a new perspective on the design and analysis of quantum algorithms, potentially unlocking significant advancements in computational complexity.
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Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.