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

Abstract

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 \( E_f^{cr}(\rho) \geq E_f^{cr}(\sigma) \) for all convex roof entanglement measures \(E_f^{cr}\) is insufficient to guarantee the existence of an LOCC transformation \(\Lambda^L(\cdot)\) from \(\rho\) to \(\sigma\); and (ii) The inequalities \(\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(\psi_l)\) for all entanglement monotones \(E\) are not sufficient to ensure the existence of an LOCC transformation from \(\{p_j, \ket{\varphi_j}\}\) to \(\{q_l, \ket{\psi_l}\}\).