Squashed Entanglement, $$\mathbf {}$$-Extendibility,Quantum Marov Chains,and Recovery Maps

Squashed entanglement (Christandl and Winter in 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 in 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 (Brandão et al. 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.