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

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