This is something that's been bothering me for a while. The way we usually first hear about causality is that "nothing travels faster than $c$". But then you learn that phase velocities can sometimes be faster than $c$, so we revise the previous statement to "information never travels faster than $c$". But maddeningly, I've never seen anyone actually define what "information" means in this context. Without a mathematical definition of information, it seems to me that the preceding statement is totally vacuous.
Can someone please provide a rigorous definition of information in this context, so that e.g. given some dynamical equations of a relativistic theory (e.g. of electrodynamics) I can verify mathematically that the equations indeed do not allow information to travel faster than light.
If this is impossible, or if nobody knows how to define information in this way, please describe the situation.
Despite many answers, nobody has yet addressed my actual question: What is a definition of information for the purposes of physics. I know about the arguments (given by people like Griffiths in his quantum mechanics book) about how certain things that appear to travel faster than light cannot be used to communicate in a way that violates causality. That is not what I'm asking! I am looking for a way to generalize the potpourri of such examples into a sharp theorem, and to that end I need a proper definition of information.
As another point for consideration, another situation in which "information" is implied to have meaningful physical interpretation is in the black hole information paradox. The rough statement of this paradox is "do black holes destroy information?". One way to interpret this question rigorously is "do black holes violate unitarity?". But what I want to know is the following: Is there a meaningful, mathematical definition of "information", which would in principle allow one to take a hypothetical theory of quantum gravity and determine rigorously whether or not black holes in that theory destroy information?
If there is no such definition of information, please provide an authoritative explanation of why not (and provide sources if possible), rather than trying to give more examples.