Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps

TL;DR

This paper explores the relationship between squashed entanglement and k-extendibility, showing that low squashed entanglement indicates proximity to highly extendible states, and discusses implications for quantum Markov chains and recovery maps.

Contribution

It provides a new proof linking small squashed entanglement to state extendibility and discusses potential generalizations to recovery maps.

Findings

Small squashed entanglement implies proximity to highly extendible states

Alternative proof of the faithfulness of squashed entanglement

Discussion of generalizations to universal recovery maps

Abstract

Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)] greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the faithfulness of squashed entanglement [Brandao, Christandl and Yard, Commun. Math. Phys. 306:805-830 (2011)]. We briefly discuss the previous and subsequent history of the Fawzi-Renner bound and related conjectures, and close by advertising a potentially far-reaching generalization to universal and functorial recovery maps for the monotonicity of the relative entropy.