>>12856256what makes information conserved in classical and quantum physics is something called “unitarity”. in classical physics a classical example is that newtonian laws (or relativistic laws” exactly specify what happens forever based on initial conditions. so if you know all the particles and fields in your system at any initial time, you can evolved it forward in time and know exactly where everything will be, and then if you then run things backward in time from there, you come exactly back to the initial conditions. in other words the information you have at any one time let’s you run the clocks forward or backward and get all the information at any other time. OTOH if it weren’t conserved, you might run the clocks back and say “heck, i can’t tell what came before this, it could be either A or B” which means even though it must HAVE been one or the other, the information about which it was somehow got lost. this doesn’t happen in classical physics.
in classical statistical mechanics the idea is that this is true too but we mere mortals can’t know everything perfectly or we can’t keep track of all the fine grained details, so the idea of entropy has to do with us losing information but still keeping a statistical handle on the macroscopic stuff we can accurately keep track of over time.
in quantum mechanics it is a bit more subtle because the wave function is probabilistic intrinsically and not due to imperfect information. so even though you might know the state perfectly, outcomes in the future are still probabilistic. but the time evolution of the state (i.e. the wave function) is unique and you still can in principle know future and past states perfectly given a known state at any time, and again there is no ambiguity where one state might rewind to an ambiguous prior state. so QM is just like classical in this respect, it’s just that the wave function is what evolves unitarily, not the canonical coordinates
