Insufficiency of pure-state ensembles in characterizing transformations of entangled states under local operations and classical communication

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 the following: (i) The conditions ${E}_{f}^{\mathrm{CR}}(\ensuremath{\rho})\ensuremath{\ge}{E}_{f}^{\mathrm{CR}}(\ensuremath{\sigma})$ for all convex roof entanglement measures ${E}_{f}^{\mathrm{CR}}$ are insufficient to guarantee the existence of an LOCC transformation ${\mathrm{\ensuremath{\Lambda}}}^{L}(\ifmmode\cdot\else\textperiodcentered\fi{})$ from $\ensuremath{\rho}$ to $\ensuremath{\sigma}$, and (ii) the inequalities ${\ensuremath{\sum}}_{j}{p}_{j}E({\ensuremath{\varphi}}_{j})\ensuremath{\ge}{\ensuremath{\sum}}_{l}{q}_{l}E({\ensuremath{\psi}}_{l})$ for all entanglement monotones $E$ are not sufficient to ensure the existence of an LOCC transformation from ${{p}_{j},|{\ensuremath{\varphi}}_{j}\ensuremath{\rangle}}$ to ${{q}_{l},|{\ensuremath{\psi}}_{l}\ensuremath{\rangle}}$.