In quantum error correction, two codes are considered to be equivalent if they differ by a locally unitary operator and a permutation. I would imagine that such codes should have the same error-correcting capabilities. Maybe it is too basic, but I am failing to see this, and I am also failing to find it explained in literature.
To make the answer easier, I am looking for the following:
1) If an error set is correctable by a code, is it also correctable by an equivalent code? And why?
2) Is this equivalence notion a natural thing to do? And why?
If it helps, feel free to restrict yourselves on stabilizer codes.
UPDATE: Why do two equivalent codes have the same error correcting capabilities?For instance, do they have the same minimum distance?