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

TL;DR

The paper tackles whether LOCC transformability of mixed entangled states can be inferred from their pure-state ensembles. It constructs counterexamples showing that ensemble-level LOCC convertibility does not imply global LOCC convertibility (Theorem 1) and demonstrates the insufficiency of both convex-roof entanglement measures (Theorem 2) and ensemble-monotone inequalities (Theorem 3) to decide LOCC transformations. By contrasting LOCC with separable operations through Lemma 1 and these theorems, the authors show that new mathematical tools beyond pure-state ensembles are necessary to fully characterize LOCC transformations of mixed states. The work clarifies fundamental limits of ensemble-based approaches and informs future directions for understanding entanglement transformations 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 \( E_f^{cr}(ρ) \geq E_f^{cr}(σ) \) for all convex roof entanglement measures \(E_f^{cr}\) is insufficient to guarantee the existence of an LOCC transformation \(Λ^L(\cdot)\) from \(ρ\) to \(σ\); and (ii) The inequalities \(\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)\) for all entanglement monotones \(E\) are not sufficient to ensure the existence of an LOCC transformation from \(\{p_j, \ket{\varphi_j}\}\) to \(\{q_l, \ket{ψ_l}\}\).