  The action of a channel on a quantum system, when non trivial, always causes
deterioration of initial quantum resources, understood as the entanglement
initially shared by the input system with some reference purifying it. One
effective way to measure such a deterioration is by measuring the loss of
coherent information, namely the difference between the initial coherent
information and the final one: such a difference is ``small'', if and only if
the action of the channel can be ``almost perfectly'' corrected with
probability one. In this work, we generalise this result to different
entanglement loss functions, notably including the entanglement of formation
loss, and prove that many inequivalent entanglement measures lead to equivalent
conditions for approximate quantum error correction. In doing this, we show how
different measures of bipartite entanglement give rise to corresponding
distance-like functions between quantum channels, and we investigate how these
induced distances are related to the cb-norm.
