A long time ago I read a newspaper article where a professor of some sort said that in the future we will be able to compress data to just two bits (or something like that).
This is of course not correct (and it could be that my memory of what he exactly stated is not correct). Understandably it would not be practical to compress any string of 0's and 1's to just two bits because (even if it was technically possible), too many different kind of strings would end up compressing to the same two bits (since we only have '01' and '10' to choose from).
Anyway, this got me thinking about the feasibility of compressing an arbitrary length string of 0's and 1's according to some scheme. For this kind of string, is there a known relationship between the string length (ratio between 0's and 1's probably does not matter) and maximum compression?
In other words, is there a way to determine what is the minimum (smallest possible) length that a string of 0's and 1's can be compressed to?
(Here I am interested in the mathematical maximum compression, not what is currently technically possible.)