  In [Gallego and Aolita, Physical Review X 5, 041008 (2015)], the authors
proposed a definition for the relative entropy of steering and showed that the
resulting quantity is a convex steering monotone. Here we advocate for a
different definition for relative entropy of steering, based on well grounded
concerns coming from quantum Shannon theory. We prove that this modified
relative entropy of steering is a convex steering monotone. Furthermore, we
establish that it is uniformly continuous and faithful, in both cases giving
quantitative bounds that should be useful in applications. We also consider a
restricted relative entropy of steering which is relevant for the case in which
the free operations in the resource theory of steering have a more restricted
form (the restricted operations could be more relevant in practical scenarios).
The restricted relative entropy of steering is convex, monotone with respect to
these restricted operations, uniformly continuous, and faithful.
