An alternate way to approach this is to use an interpretation which does not require collapse nor non-determinism. All of the interpretations are simply ways to reconcile the mathematics of a quantum reality with the mathematics of a classical reality as we observe it. There is no wave function collapse in quantum mechanics proper -- it is something which appears in the most common interpretation, the Copenhagen interpretation.
We could use other interpretations to explore this answer. Pilot wave comes to mind as an excellent example. In the pilot wave interpretation, we can measure the state of particles that are constantly being affected by a "pilot wave," a wave function which jostles the particles, changing their state. Like all interpretations of QM, this view is perfectly consistent with the fundamental equations of QM. However, instead of a wave function collapse, like the Copenhagen Interpretation has, we have a pilot wave.
The tricky bit about this pilot wave is it's equation at every moment in time is dependent on the state of all particles, at that moment, even those which are remote. This weirdness is how pilot wave gets around classical behaviors -- it has a wave that propagates infinitely fast. It can be shown that this yields the same statistical results that we get from the Copenhagen interpretation, with its wave function collapse, but no collapse is required.
In this, we find it trivial to show that information is conserved for all actions, even "measurements," because the pilot wave gets defined with respect to the unital operators we see in quantum mechanics. However, that information has been dispersed across every particle in the known universe.
So it shows that, by that interpretation, information is conserved across the entire universe, but any sub-system within the universe will lose information as it is scattered to all of the particles in existence.