  In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum
error correcting protocols designed for tolerance against qubit loss, it was
shown that these protocols have the undesirable effect of magnifying the
effects of depolarization noise. This raises the question of which general
properties of quantum error-correcting codes might explain such an apparent
trade-off between tolerance to located and unlocated error types. We extend the
counting argument behind the well-known quantum Hamming bound to derive a bound
on the weights of combinations of located and unlocated errors which are
correctable by nondegenerate quantum codes. Numerical results show that the
bound gives an excellent prediction to which combinations of unlocated and
located errors can be corrected with high probability by certain large
degenerate codes. The numerical results are explained partly by showing that
the generalized bound, like the original, is closely connected to the
information-theoretic quantity the quantum coherent information. However, we
also show that as a measure of the exact performance of quantum codes, our
generalized Hamming bound is provably far from tight.
