It is a bit more difficult to see this because you are describing a continuous time process, however, the answer is that yes you would be able to to figure out

It is a bit more difficult to see this because you are describing a continuous time process, however, the answer is that yes you would be able to to figure out the time t1 given you knew had a snapshot of the whole system through the time evolution equation. Perhaps your confusion stems from the fact that the time t2 is observed, and it seems like you can't go back after that. An easier to grasp manifestation of this problem is that if you find Schrodinger's cat dead you can't tell if was always dead or if was dead only after your experiment. The reason is because that's only with the narrow knowledge of that cat being dead (maybe the would call this the cat state subsystem). It might be unsatisfactory to you, but in the theory of quantum mechanics, if you knew the state of everything (say the room, measurement apparatus, and so on), then you would be able to trace back to whether the cat was alive or dead. The moral is that the measurement looks irreversible if you look only at the system you are observing, but overall, people accept that there is a overarching greater space (the "environment, universe" where everything is just a unitary operation in motion according to the time evolution equation [1]). To be clear, the time evolution equation only admits reversible solutions. So with full knowledge, of say, the room and measurement apparatus, it would be possible to deduce the original state of the cat. https://www.quantiki.org/wiki/church-larger-hilbert-space https://www.quantiki.org/wiki/church-larger-hilbert-space