Firstly, I am a layman with an interest in physics only so please forgive any ignorance I demonstrate here. This question puzzles me and I'm sure there's a good answer. I've heard it said repeatedly when referencing things like the black hole information paradox that information is preserved. This was the impetus for Hawking and his theory on radiating black holes (Edit here - I believe I'm wrong about this in particular, however the two are related regardless of if evaporating black holes are the source of or part of the solution to the information paradox). The general description one gets here is that information is seemingly destroyed when it falls into a black hole, which violates the conservation of information. It's stated in simple terms like the following:
If one were to be able to track the position direction and velocity of every particle in a building, which is blown up, one could effectively run the clock backwards and reconstruct the building.
This seems straightforward enough. Given an input and a deterministic system, you can simply undo what was done using the same rules, but it then clashes with the idea of the inherently uncertain probabilistic nature of the atomic scale. We know this uncertainly exists, and we can see the effects of quantum uncertainly manifest in real world randomness, consider radioactive decay or interference patterns and so on. How then are these two ideas reconcilable? If I can not be certain of the all of the descriptors of a given particle, or indeed get the same numbers twice in a row when measuring location or momentum as it is forbidden by nature itself, I could never perform an experiment which is the equivalent of the aforementioned exploding building. I understand that on the macro scale, things seem predictable and deterministic, presumably because the average of the uncertainty gives the appearance of a deterministic system on a macro scale, or the determinism is otherwise emergent... but if the reality of the situation is that a given particle only has a chance of being at any given location with a specific momentum, it logically follows that one could never reconstruct any object after a state change with absolute certainty. I can see getting very close, but that doesn't really seem to constitute a 'law', like the conservation of information is said to be. What am I missing here?