  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.
