Insufficiency of Pure-State Ensembles in Characterizing Transformations of Entangled States under LOCC

The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions $E_f^{cr}(\rho )\ge E_f^{cr}(\sigma )$ for all convex roof entanglement measures $E_f^{cr}$ is insufficient to guarantee the existence of an LOCC transformation ${\mathrm{\Lambda }}^L(\cdot )$ from $\rho$ to $\sigma$; and (ii) The inequalities ${\sum }_j{p}_{j}E({\phi }_j)\ge {\sum }_l{q}_{l}E({\psi }_l)$ for all entanglement monotones $E$ are not sufficient to ensure the existence of an LOCC transformation from $\{p_j,\mathrm{\mid }{\phi }_j⟩\}$ to $\{q_l,\mathrm{\mid }{\psi }_l⟩\}$.