In a quantum double slit experiment, one starts at t0 with a wave function that propagates through two slits, interferes, and probabilities for various positions at the final stage at t1 can be computed. This is a physical representation that can be endorsed by an observer at the beginning of the experiment and allows her to predict possible outcomes. However at the end of the experiment, an observer has observed a definite outcome corresponding to a precise position (in Copenhagen parlance, there's a "collapse"). So in principle, this final observer could represent what has just happened differently: have a "collapsed" wave function with well defined position (up to a certain precision maybe) at t1 and apply the schrödinger equation backward in time to see what the wave function was at t0. Then the result would be different: let's call psi0(t) the wave function of the initial observer and psi1(t) that of the final observer.
Whether this kind of backward in time calculation is legitimate, meaningful, useful or anything is not my concern (it is feasible so let's assume it is legitimate for the sake of the question). My question is: how would the two wave functions psi0 and psi1 be related? It seems that the final observer has "more information" than the initial observer: does it translate in one way or another? Is there an asymmetry between psi0 and psi1, an objective way to tell that psi1 contains more information about the world?
What I find puzzling is that any wave function can be represented as a vector in Hilbert space, and I can't see how a vector could be said to carry more "information" than another. They're just vectors. On the other hand, it seems obvious that the final observer has more information than the initial one, since she has one more measurement record.