Title: Non-Commutative Differential Geometry for Quantum State Decomposition

Jincheng Zhang

Published Sep 6, 2026• 0 citations

Abstract

Quantum state decomposition is a fundamental problem in quantum mechanics, requiring precise and efficient representations of complex quantum states. Traditional methods often struggle with the inherent complexities of these states, particularly in dealing with multiple possible states and their interactions. This paper explores the application of non-commutative geometry as a novel framework for quantum state decomposition, proposing a new set of geometric predicates and operations designed to enhance the representation of quantum states. We introduce a new formalism that leverages the non-commutative nature of quantum spaces to simplify calculations and improve the ability to analyze quantum systems. The core mechanism centers around the concept of 'geometric predicates' which dynamically reshape the quantum state space, allowing for a more expressive and potentially reduced number of degrees of freedom. This approach offers a significant departure from conventional methods, potentially yielding more insightful and efficient results. We present a preliminary investigation, demonstrating the efficacy of this framework through illustrative examples and exploring its potential for broader applicability.

DOI: 10.5281/zenodo.22497083

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