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

TL;DR

This paper demonstrates that pure-state ensembles are insufficient for characterizing LOCC transformations of mixed entangled states, addressing open questions about entanglement measures and transformation conditions.

Contribution

It proves that ensemble-based entanglement conditions do not fully determine LOCC convertibility of mixed states, highlighting limitations of pure-state ensemble approaches.

Findings

Pure-state ensemble conditions are insufficient for LOCC transformations.

Entanglement measures inequalities do not guarantee LOCC convertibility.

Addresses open questions in entanglement transformation theory.

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 \(\{q_l, \ket{\psi_l}\}\).