To answer the question we should define what is a quantum system, information, measure,..But let us try to be more general.

If the system is represented by variables X (maps) belonging to some infinite dimensional vector space, then if it meets some general conditions, one can prove that the set of states (meaning all possible values of X) is isomorphic to an open subset O of a Hilbert space. So one can say that O represents the total sum of possible information. But because the the system usually follows some physical rules, its state would be only one vector S of O. Measures are done by getting a finite number of data, so we have only an approximation of this point : the observables define a neighborhood T, contained in O, which contains S.

So the process of measurement can be assessed, from the information point of view, by the size of T..

And one can see that the problemis actually how much information one can extract from a given system. This is similar to the entropy . It depends of the observables (how many data one collects) and of the system (the size of O).

If the original system is in the statical state defined by \rho_{pre}, and after measurement it is in the state \rho_{post}, then IIRC the information gain is 

D(\rho_{post} | \rho_{pre}) =

tr(x \log(x)|_{x  = \rho_{post}\rho_{pre}^{-1}} \rho_{pre})

(here we only have to define the inverse on the orthogonal complement of the kernel of \rho_{pre})