Can Entanglement Measures Predict LOCC Majorization Ordering?  A Machine Learning Study of Two-Qubit Systems

Majorization ordering provides a fundamental framework for understanding LOCC (Local Operations and Classical Communication) transformations in quantum systems, but its relationship to standard entanglement monotones remains unclear. Using direct eigenvalue-based majorization testing on 1000 two-qubit entangled states and 50,000 pairwise comparisons, we investigate whether entanglement measures can predict majorization comparability. While individual monotones achieve only 52\% AUC (Area Under Curve), combined machine learning models (Random Forest) achieve 71\% AUC. Critically, information-geometric metrics (Bures and Hilbert-Schmidt distances) predict majorization 4.6× better than entanglement monotones, revealing that majorization is fundamentally a \emph{geometric property} of quantum state space. We find a counterintuitive pattern: incomparable state pairs exhibit \emph{lower} average entanglement measure differences than comparable pairs, demonstrating majorization incomparability is orthogonal to entanglement magnitude. \textbf{Key contribution:} This provides the first quantitative benchmark connecting resource-theoretic ordering (majorization) with machine learning prediction, establishing majorization incomparability as a geometric phenomenon independent of entanglement measures. We identify key predictive features and discuss implications for quantum resource theory and LOCC protocols.