This got me to some interesting thinking. If the library contains no information because you need the information you look for, what about the ability of it to at least match to information you look for? Or put another way, the library does begin to have information if you have the information you're looking for. The fact of finding the particular information is different than the library not containing it.
I can't seem to figure out how to type this out in a way that maks sense but basically I'm thinking when an AI like GPT-3 is working its sort of sorting through the library of babel and finding words. Or when speaking its as though the library of babel is at immediate call in the brain, which sorts through near instantly finding the book that satisfies the next word. The website that allows browsing the library helps show what I mean, you can look on it and click random and search for information in it. The thing itself contains "no information" but it also does as in this case you may find something (first page I saw had the word 'beef')