Additive entanglemement measures cannot be more than asymptotically continuous

TL;DR

This paper proves that additive entanglement measures invariant under permutations or local unitaries cannot exhibit more than asymptotic continuity, highlighting fundamental limitations in quantifying entanglement.

Contribution

It establishes a theoretical limitation on additive entanglement measures, extending previous results to a broader class of measures and invariances.

Findings

Additive measures cannot be more than asymptotically continuous.

The proof generalizes a protocol for embezzling entanglement.

Results apply to measures invariant under local unitaries.

Abstract

In this short note, we show that any non-constant quantity defined on density matrices that is additive on tensor products and invariant under permutations cannot be "more than asymptotically continuous." The proof can be adapted to show that any additive entanglement measure (on any number of parties) that is invariant under local unitary operations also cannot be more than asymptotically continuous. The proof is a direct consequence of generalizing a protocol in arXiv:0804.4118 for embezzling entanglement.