  We prove that if any error channel has a Kraus decomposition that is
simultaneously correctable and Hilbert-Schmidt (HS) complete, then the
existence of Kraus sets with these properties guarantees the correctability of
all quantum channels. As a proof of the existence of such Kraus sets, the
$n$-level depolarization channel is shown to have a random-unitary (RU)
decomposition that is both HS complete and correctable due its RU nature,
thereby proving that all quantum channels are correctable. As an application,
conditions for universal error-correction operations are presented.
