Robust generalized quantum Stein's lemma

TL;DR

This paper proves the robustness of the generalized quantum Stein's lemma, extending its applicability from iid states to almost-iid states using a new continuity bound and asymptotic closeness in quantum Wasserstein distance.

Contribution

It establishes the robustness of the generalized quantum Stein's lemma for almost-iid states, correcting a logical gap in previous proofs.

Findings

The lemma holds for almost-iid states, not just iid states.

A new continuity bound for the relative entropy of entanglement is introduced.

Almost-iid states and iid states are asymptotically close in quantum Wasserstein distance.

Abstract

The generalized quantum Stein's lemma provides an explicit expression for the optimal error exponent when distinguishing many independent and identically distributed (iid) copies of a given bipartite state from the set of separable bipartite states. Here we prove that this result is robust, in the sense that the iid assumption can be relaxed to almost-iid. In particular, our result shows that the original argument of Brand\~ao and Plenio, which contains a logical gap, can be made rigorous. Our proof relies on a novel continuity bound for the relative entropy of entanglement with respect to the quantum Wasserstein distance. Combined with a recent insight that almost-iid states and their exact iid counterparts are asymptotically close in this distance, the bound implies that their relative entropies of entanglement coincide asymptotically.