Entanglement-measure vectors with generalized entanglement axioms

Despite significant advances in quantum information theory, the well-known entanglement axiom cannot be used to completely distinguish between different types of entanglement. We address this fundamental limitation by proposing generalized entanglement axioms for the complete characterization of entanglement. We firstly extend entanglement axioms under vector majorization---a mathematical framework that extends beyond conventional scalar measures. We then introduce entanglement-measure vectors that enable comprehensive classification of all bipartite pure states under local unitary operations. We further demonstrate how this framework naturally extends to multipartite systems through marginal information analysis. This work establishes a unified, mathematically rigorous methodology for entanglement classification with implications for both quantum foundations and quantum technologies.