  We consider the problem of correcting the errors incurred from sending
classical or quantum information through a noisy quantum environment by schemes
using classical information obtained from a measurement on the environment. We
give a conditions for quantum or classical information (prepared in a specified
input basis B) to be corrigible based on a measurement M. Based on these
criteria we give examples of noisy channels such that (1) no information can be
corrected by such a scheme (2) for some basis B there is a correcting
measurement M (3) for all bases B there is an M (4) there is a measurement M
which allows perfect correction for all bases B. The last case is equivalent to
the possibility of correcting quantum information, and turns out to be
equivalent to the channel allowing a representation as a convex combination of
isometric channels. Such channels are doubly stochastic but not conversely.
