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This research establishes a rigorous mathematical equivalence between quantum probability on metric graphs and geometric curvature derived from Wilson lines and holonomy. By reinterpreting the graph trace formula through von-Neumann's ergodic theorem, the study unifies spectral analysis of networks with fundamental principles of quantum measurement. This theoretical advance provides a new framework for quantifying connections between quantum information theory and network geometry without relying on mere analogy.

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Graph Trace Formula Links Quantum Probability to Curvature

Quantum ZeitgeistAugust 1, 20265 min read

For a metric graph equipped with the Laplacian operator, the graph trace formula admits a new interpretation in terms of quantum probability and curvature. This approach is based on a notion of graph curvature inspired by Wilson-lines and holonomy, together with von-Neumann’s ergodic theorem.

A connection between quantum probability and the geometry of networks has emerged from a reinterpretation of the graph trace formula, decades after its initial formulation. Tianhong Zhao of University of Science and Technology of China demonstrates that this formula, which links a graph’s spectrum to its primitive closed paths, can be understood through the lens of quantum measurement and curvature. An early trace formula for the Laplacian on metric graphs was first derived by Roth in 1984, with subsequent advancements including a 1999 scattering-theoretic approach by Kottos and Smilansky relating the quantum…

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