Product-State Manifolds for M Quantum Systems with N Levels using the Fano form and the Induced Euclidean Metric

• URL: http://arxiv.org/abs/2509.02891v1
• Date: Tue, 02 Sep 2025 23:31:07 GMT
• Title: Product-State Manifolds for M Quantum Systems with N Levels using the Fano form and the Induced Euclidean Metric
• Authors: Fotios D. Oikonomou,
• Abstract summary: In quantum mechanics separable states can be characterized as convex combinations of product states whereas non-separable states exhibit entanglement.<n>In this study, we explore the connection between product states and geometric structures.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In quantum mechanics separable states can be characterized as convex combinations of product states whereas non-separable states exhibit entanglement. Quantum entanglement has played a pivotal role in both theoretical investigations and practical applications within quantum information science. In this study, we explore the connection between product states and geometric structures, specifically manifolds and their associated geometric properties such as the first fundamental form (metric). We focus on the manifolds formed by the product states of M systems of N levels, examining the induced metric derived from the Euclidean metric. For elementary cases we will compute the Levi-Civita connection, and, where computationally tractable, the scalar curvature.

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