A Foundational Protocol for Reproducible Visualization in Multivariate Quantum Data

The visualization of high-dimensional data is a cornerstone of modern scientific inquiry, particularly in quantum physics, where complex non-linear interactions define system behavior. While linear dimensionality reduction methods provide mathematical guarantees of reproducibility, they fail to capture the intricate manifolds underlying such data. Non-linear techniques like Uniform Manifold Approximation and Projection (UMAP) are therefore essential, but their stochastic optimization introduces a fundamental challenge: the lack of reproducibility across independent runs. In this work, we introduce a foundational protocol to establish UMAP as a reproducible tool for scientific visualization. We define explicit, quantitative criteria for embedding convergence, requiring that repeated executions of UMAP under fixed parameters consistently produce a single connected embedding with zero variance in the number of connected components. This criterion transforms UMAP from an exploratory heuristic into a deterministic mapping procedure. Applying the protocol to high-dimensional multivariate quantum data, we demonstrate that feature standardization promotes rapid and consistent convergence at substantially smaller neighborhood sizes, whereas raw data require careful parameter tuning to achieve reproducibility. Our framework provides a rigorous methodological foundation for distinguishing robust visual structures from stochastic artifacts, elevating non-linear visualization to a reproducible component of the scientific process.