Detecting entanglement in high-spin quantum systems via a stacking ensemble of machine learning models

Reliable quantification of quantum entanglement in high-spin or many body systems remains a major computational challenge. Extending machine learning techniques to genuinely high dimensional settings is urgently needed. In this study, we investigate ensemble machine learning as a scalable framework for estimating entanglement, quantified by the negativity, in high-spin quantum systems. We construct a stacked ensemble regressor integrating Neural Networks, XGBoost, and Extra Trees. The model is trained on real-coefficient pure states and mixed Werner states for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J = 1/2,\, 1$$\end{document}, and 5, corresponding to the orthogonal ensemble characteristic of time-reversal-symmetric systems. With CatBoost serving as the meta-learner, the ensemble achieves consistently high predictive accuracy. Statistical validation across five independent random seeds confirms negligible run-to-run variance in all reported metrics. Residual analysis reveals a heteroscedastic error structure: prediction fidelity is highest near \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N} \approx 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N} \approx 1$$\end{document}, with peak variance in the intermediate regime (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N} \approx 0.4$$\end{document}–0.7). Empirical scaling laws for training time and memory consumption are derived, showing that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2J+1)^{4}$$\end{document} growth of the Werner-state feature space poses a scalability ceiling for raw density-matrix representations beyond \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J = 5$$\end{document}. Moreover, we derive an empirical formula linking the required dataset size to system dimensionality and desired prediction accuracy. Our findings demonstrate that ensemble learning provides a robust and trustworthy tool for characterizing entanglement in high dimensional quantum physics.