<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(J = 1/2,\, 1\)</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {N} \approx 0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {N} \approx 1\)</EquationSource> </InlineEquation>, with peak variance in the intermediate regime (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {N} \approx 0.4\)</EquationSource> </InlineEquation>–0.7). Empirical scaling laws for training time and memory consumption are derived, showing that the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((2J+1)^{4}\)</EquationSource> </InlineEquation> growth of the Werner-state feature space poses a scalability ceiling for raw density-matrix representations beyond <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(J = 5\)</EquationSource> </InlineEquation>. 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.</p>

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

  • M. Y. Abd-Rabbou,
  • Ahmed A. Zahia,
  • Amr M. Abdallah,
  • Ashraf A. Gouda,
  • Cong-Feng Qiao

摘要

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 \(J = 1/2,\, 1\) , 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 \(\mathcal {N} \approx 0\) and \(\mathcal {N} \approx 1\) , with peak variance in the intermediate regime ( \(\mathcal {N} \approx 0.4\) –0.7). Empirical scaling laws for training time and memory consumption are derived, showing that the \((2J+1)^{4}\) growth of the Werner-state feature space poses a scalability ceiling for raw density-matrix representations beyond \(J = 5\) . 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.