It's been shown that black holes don't destroy information. There's a little-known scenario in computer science (Specifically AI) where an advanced civilization creates an artificial intelligence that in its quest to improve itself, storing more data and computing at ever increasing densities eventually becomes so massive and dense that it creates a black hole. If this were to happen and a black hole were formed, would it be possible for the system to maintain high enough levels of data entropy? In this scenario it's also sometimes imagined that the black hole might be used to communicate with other entities that have accomplished the same thing.

I ask because in this situation I can imagine the system being destroyed by a black hole but remembering that black holes don't destroy information it seems that it could be possible for the system to remain intact. We could also assume that the system has moved beyond computation using silicon-based chips or similar to some materials capable of greater density.