Well, let's say you measured the times at which the bottom pendulum crossed the median. You would get a list of times that were spaced here and there, some near others, some far. This would be one possible set of information that the pendulum produces.

The idea is that, presumably, this list of times "looks" very random and can't be expressed by a much shorter, compressed, set of data, not even by providing the pendulum equation, and initial conditions with finite precision. (You could, if it weren't chaotic, just a single pendulum.) This would mean it has high entropy content, in the information theory sense. (I don't actually know, since I haven't done the experiment nor read about it being done, but it's a reasonable guess.)

My thoughts exactly. I don't know a whole lot about information theory or quantum mechanics, but my PhD is in dynamical systems and I've never heard tem described as 'random' or 'information producing'.