There is a very easy, but highly unsatisfying way that your question could be answered:
Given a noisy channel, a code is error-correcting, iff it increases the capacity. Normally, the idea is that the channel has zero-capacity and afterwards, it has capacity.
Now, the various capacities (e.g. classical, quantum-classical, quantum, one-shot-quantum) are usually expressed via entropic quantities (e.g. Shannon's noisy channel theorem, the LSD-theorem), which seems to answer the question.
However, obviously, the requirement of a quantum channel to be error-correcting, if phrased this way, is very broad and usually very hard to deal with. In essence, the whole point of the subfield of error-correction is to find better ways to express this. I do have some ideas of where to look for more connections between entropy and error-correcting abilities, but I have only ever heard them floating around and I am more on the quantum Shannon than the error-correcting side of QI. One principle would be that a noisy channel typically increases the entropy of the input state, i.e. to obtain error-correcting-codes, one would have to "get rid" of the additional entropy by concentrating them in some redundant part of the code. But I have never seen this fleshed out a lot. There is a paper, which might use this concept, but I haven't really looked at it and I don't know the author, so I can't say anything about it: http://arxiv.org/abs/quant-ph/0703258.